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Integrals

Class 12th Mathematics Part Ii Bihar Board Solution
Exercise 7.1
  1. sin 2x Find an anti-derivative (or integral) of the following functions by the…
  2. cos 3x Find an anti-derivative (or integral) of the following functions by the…
  3. e2x Find an anti-derivative (or integral) of the following functions by the…
  4. (ax + b)^2 Find an anti-derivative (or integral) of the following functions by…
  5. sin 2x - 4 e3x Find an anti-derivative (or integral) of the following functions…
  6. Find the following integrals. integrate (4e^3x + 1) dx
  7. integrate x^2 (1- 1/x^2) dx Find the following integrals.
  8. integrate (ax^2 + bx+c) dx Find the following integrals.
  9. integrate 2x^2 + e^x dx Find the following integrals.
  10. integrate (root x - 1/root x)^2 dx Find the following integrals.
  11. integrate x^3 + 5x^2 - 4/x^2 dx Find the following integrals.
  12. integrate x^3 + 3x+4/root x dx Find the following integrals.
  13. integrate x^3 - x^2 + x-1/x-1dx Find the following integrals.
  14. integrate (1-x) root xdx Find the following integrals.
  15. integrate root x (3x^2 + 2x+3) dx Find the following integrals.
  16. integrate (2x-3cosx+e^x) dx Find the following integrals.
  17. integrate (2x^2 - 3sinx+5 root x) dx Find the following integrals.…
  18. integrate secx (secx+tanx) dx Find the following integrals.
  19. integrate sec^2x/cosec^2xdx Find the following integrals.
  20. integrate 2-3sinx/cos^2xdx Find the following integrals.
  21. The anti-derivative of (root x + 1/root x) equalsA. 1/3 x^1/3 + 2x^1/2 + c B.…
  22. If d/dx f (x) = 4x^3 - 3/x^4 such that f(2) = 0. Then f(x) isA. x^4 + 1/x^3 -…
Exercise 7.10
  1. Evaluate the integrals using substitution. integrate _0^1 x/x^2 + 1 dx…
  2. integrate _0^pi /2 root sinphi cos^5phi d phi Evaluate the integrals using…
  3. integrate _0^1sin^-1 (2x/1+x^2) dx Evaluate the integrals using substitution.…
  4. integrate _0^2root x+2 (putx+2 = t^2) Evaluate the integrals using substitution.…
  5. integrate _0^pi /2 sinx/1+cos^2xdx Evaluate the integrals using substitution.…
  6. integrate _0^2 dx/x+4-x^2 Evaluate the integrals using substitution.…
  7. integrate _-1^1 dx/x^2 + 2x+5 Evaluate the integrals using substitution.…
  8. integrate _1^2 (1/x - 1/2x^2) e^2x dx Evaluate the integrals using substitution.…
  9. The value of the integral integrate _ 1/3^1 (x-x^3)^1/3/x^4 dx isA. 6 B. 0 C. 3…
  10. If f (x) = integrate _0^xtsintegrate dt then f'(x) isA. B. xsinx C. xcosx D.…
Exercise 7.11
  1. integrate _0^pi /2 cos^2xdx By using the properties of definite integrals,…
  2. integrate _0^pi /2 root sinx/root sinx + root cosx dx By using the properties of…
  3. integrate _0^pi /2 sin^3/2 xdx/sin^3/2 x+cos^3/2 x dx By using the properties of…
  4. integrate _0^pi /2 cos^5xdx/sin^5x+cos^5x By using the properties of definite…
  5. integrate _-5^5 |x+2|dx By using the properties of definite integrals, evaluate…
  6. integrate _2^8 |x-5|dx By using the properties of definite integrals, evaluate…
  7. integrate _0^1z (1-x)^n dx By using the properties of definite integrals,…
  8. integrate _0^pi /4 log (1+tanx) dx By using the properties of definite…
  9. integrate _0^2x root 2-xdx By using the properties of definite integrals,…
  10. integrate _0^pi /2 (2logsinx-logsin2x) dx By using the properties of definite…
  11. integrate _ - pi /2^pi /2 sin^2xdx By using the properties of definite…
  12. integrate _0^pi xdx/1+sinx By using the properties of definite integrals,…
  13. integrate _ - pi /2^pi /2 sin^7xdx By using the properties of definite…
  14. By using the properties of definite integrals, evaluate the integrals integrate…
  15. integrate _0^pi /2 sinx-cosx/1+sinxcosxdx By using the properties of definite…
  16. integrate _0^pi log (1+cosx) dx By using the properties of definite integrals,…
  17. integrate _0^a root x/root x + root a-x dx By using the properties of definite…
  18. integrate _0^4 |x-1|dx By using the properties of definite integrals, evaluate…
  19. Show that integrate _0^af (x) g (x) dx = 2 integrate _0^af (x) dx_s if f and g…
  20. The value of integrate _ - pi /2^pi /2 (x^3 + xcosx+tan^3x+1) dx A. 0 B. 2 C.…
  21. The value of integrate _0^pi /2 log (4+3sinx/4+3cosx) dxjs A. 2 B. 3/4 C. 0 D.…
Miscellaneous Exercise
  1. Integrate the function: 1/x-x^3
  2. Integrate the function: 1/root x+a + root x+b
  3. Integrate the function: 1/x root ax-x^2 [: putx = a/t]
  4. Integrate the function: 1/x^2 (x^4 + 1)^3/4
  5. Integrate the function: 1/x^1/2 + x^1/3 [hintegrate 1/x^1/2 + x^1/3 = 1/x^1/3…
  6. Integrate the function: 5x/(x+1) (x^2 + 9)
  7. Integrate the function: sinx/sin (x-a)
  8. Integrate the function: e^5logx-e^4logx/e^3logx-e^2logx
  9. Integrate the function: cosx/root 4-sin^2x
  10. Integrate the function:
  11. Integrate the function: 1/cos (x+a) cos (x+b)
  12. Integrate the function: x^3/root 1-x^8
  13. Integrate the function: e^x/(1+e^x) (2+e^x)
  14. Integrate the function: 1/(x^2 + 1) (x^2 + 4)
  15. Integrate the function: cos^3xe^logsinx
  16. Integrate the function: e^3logx (x^4 + 1)^-1
  17. Integrate the function: f^there there eξ sts (ax+b) [f (ax+b)]^n
  18. Integrate the function: 1/root sin^3xsin (x + alpha)
  19. Integrate the function: sin^-1root x-cos^-1root x/sin^-1root x+cos^-1root x , x…
  20. Integrate the function: root 1 - root x/1 + root x
  21. Integrate the function: 2+sin2x/1+cos2xe^x
  22. Integrate the function: x^2 + x+1/(x+1)^2 (x+2)
  23. Integrate the function: tan^-1root 1-x/1+x
  24. Integrate:
  25. Evaluate the definite integral: integrate _ pi /2^pi e^x (1-sinx/1-cosx) dx…
  26. Evaluate the definite integral: integrate _0^pi /4 sinxcosx/cos^4x+sin^4xdx…
  27. Evaluate the definite integral: integrate _0^pi /2 cos^2xdx/cos^2x+4sin^2x…
  28. Evaluate the definite integral: integrate _ pi /6^pi /3 sinx+cosx/root sin2x dx…
  29. Evaluate the definite integral: integrate _0^1 dx/root 1+x - root x…
  30. Evaluate the definite integral: integrate _0^pi /4 sinx+cosx/9+16sin2xdx…
  31. Evaluate the definite integral: integrate _0^pi /2 sin2xtan^-1 (sinx) dx…
  32. Evaluate the definite integral: integrate _0^pi xtanx/secx+tanxdx…
  33. Evaluate the definite integral: integrate _1^4 [|x-1|+|x-2|+|x-3|]dx…
  34. Prove: integrate _1^3 dx/x^2 (x+1) = 2/3 + log 2/3
  35. Prove: integrate _0^1xe^x dx = 1
  36. Prove: integrate _-1^1x^17cos^4xdx = 0
  37. Prove: integrate _0^pi /2 sin^3xdx = 2/3
  38. Prove: integrate _0^pi /4 2tan^3xdx = 1-log2
  39. Prove: integrate _0^1sin^-1xdx = pi /2 - 1
  40. Evaluate integrate _0^1e^2-3x dx as a limit of a sum.
  41. integrate dx/e^x + e^-x is equal to Choose the correct answersA. log (e^x -…
  42. integrate cos2x/(sinx+cosx)^2 dx is equal to Choose the correct answersA.…
  43. If f (a + b - x) = f (x), then integrate _a^bxf (x) is equal to Choose the…
  44. The value of integrate _0^1tan^-1 (2x-1/1+x-x^2) dx Choose the correct…
Exercise 7.2
  1. 2x/1+x^2 Integrate the functions.
  2. (logx)^2/x Integrate the functions.
  3. 1/x+xlogx Integrate the functions.
  4. sin x sin (cos x) Integrate the functions.
  5. Integrate the functions. sin (ax + b) cos (ax + b)
  6. root ax+b Integrate the functions.
  7. x root x+2 Integrate the functions.
  8. x root 1+2x^2 Integrate the functions.
  9. (4x+2) root x^2 +x+1 Integrate the functions.
  10. 1/x - root x Integrate the functions.
  11. x/root x+4 , x0 Integrate the functions.
  12. (x^3 - 1)^1/x x^5 Integrate the functions.
  13. x^2/(2+3x^3)^3 Integrate the functions.
  14. 1/x (logx)^m , x0 , m not equal 1 Integrate the functions.
  15. x/9-4x^2 Integrate the functions.
  16. e^2x+3 Integrate the functions.
  17. x/e^x^2 Integrate the functions.
  18. e^tan^-1x/1+x^2 Integrate the functions.
  19. e^2x - 1/e^2x + 1 Integrate the functions.
  20. e^2x - e^-2x/e^2x + e^-2x Integrate the functions.
  21. tan^2 (2x - 3) Integrate the functions.
  22. sec^2 (7 - 4x) Integrate the functions.
  23. sin^-1x/root 1-x^2 Integrate the functions.
  24. 2cosx-3sinx/6cosx+4sinx Integrate the functions.
  25. 1/cos^2x (1-tanx)^2 Integrate the functions.
  26. cosroot x/root x Integrate the functions.
  27. root sin2x cos2x Integrate the functions.
  28. cosx/root 1+sinx Integrate the functions.
  29. cot x log sin x Integrate the functions.
  30. sinx/1+cosx Integrate the functions.
  31. sinx/(1+cosx)^2 Integrate the functions.
  32. 1/1+cotx Integrate the functions.
  33. 1/1-tanx Integrate the functions.
  34. root tanx/sinxcosx Integrate the functions.
  35. (1+logx)^2/x Integrate the functions.
  36. (x+1) (x+logx)^2/x Integrate the functions.
  37. x^3sin (tan^-1x^4)/1+x^8 Integrate the functions.
  38. Choose the correct answer: A. 10x x^10 + C B. 10x + x^10 + C C. (10x x^10)1 + C…
  39. integrate dx/sin^2xcos^2x equalsA. tan x + cot x + C B. tan x - cot x + C C.…
Exercise 7.3
  1. sin^2 (2x+5) Find the integrals of the functions.
  2. cos2xcos4xcos6x Find the integrals of the functions.
  3. sin3xcos4x Find the integrals of the functions.
  4. sin^3 (2x+1) Find the integrals of the functions.
  5. sin^3xcos^3x Find the integrals of the functions.
  6. sinxsin2xsin3x Find the integrals of the functions.
  7. sin4xsin8x Find the integrals of the functions.
  8. 1-cos/1+cos Find the integrals of the functions.
  9. cosx/1+cosx Find the integrals of the functions.
  10. sin^4x Find the integrals of the functions.
  11. cos^42x Find the integrals of the functions.
  12. sin^2x/1+cosx Find the integrals of the functions.
  13. cos2x-cos2alpha /cosx-cosalpha Find the integrals of the functions.…
  14. cosx-sinx/1+sin2x Find the integrals of the functions.
  15. tan^32xsec2x Find the integrals of the functions.
  16. tan^4x Find the integrals of the functions.
  17. sin^3x+cos^3x/sin^2xcos^2x Find the integrals of the functions.
  18. cos2x+2sin^2x/cos^2x Find the integrals of the functions.
  19. 1/sinxcos^3x Find the integrals of the functions.
  20. cos2x/(cosx+sinx)^2 Find the integrals of the functions.
  21. sin^-1 (cosx) Find the integrals of the functions.
  22. 1/cos (x-a) cos (x-b) Find the integrals of the functions.
  23. integrate sin^2x-cos^2x/sin^2xcos^2xdx A. tanx+cotx+c B. tanx+cosecx+c C.…
  24. integrate e^x (1+x)/cos^2 (e^xx) A. -cot (ex^4) + c B. tan (e^x) + c C. tan…
Exercise 7.4
  1. 3x^2/x^6 + 1 Integrate the functions.
  2. 1/root 1+4x^2 Integrate the functions.
  3. 1/root (2-x)^2 + 1 Integrate the functions.
  4. 1/root 9-25x^2 Integrate the functions.
  5. 3x/1+2x^4 Integrate the functions.
  6. x^2/1-x^6 Integrate the functions.
  7. x-1/root x^2 - 1 Integrate the functions.
  8. x^2/root x^6 + a^6 Integrate the functions.
  9. sec^2x/root tan^2x+4 Integrate the functions.
  10. 1/root x^2 + 2x+2 Integrate the functions.
  11. 1/9x^2 + 6x+5 Integrate the functions.
  12. 1/root 7-6x-x^2 Integrate the functions.
  13. 1/root (x-1) (x-2) Integrate the functions.
  14. 1/root 8+3x-x^2 Integrate the functions.
  15. 1/root (x-a) (x-b) Integrate the functions.
  16. 4x+1/root 2x^2 + x-3 Integrate the functions.
  17. x+2/root x^2 - 1 Integrate the functions.
  18. 5x-2/1+2x+3x^2 Integrate the functions.
  19. 6x+7/root (x-5) (x-4) Integrate the functions.
  20. x+2/root 4x-x^2 Integrate the functions.
  21. x+2/root x^2 + 2x+3 Integrate the functions.
  22. x+3/x^2 - 2x-5 Integrate the functions.
  23. 5x+3/root x^2 + 4x+10 Integrate the functions.
  24. Choose the correct answer: integrate dx/x^2 + 2x+2 A. xtan^-1 (x+1) + c B.…
  25. Choose the correct answer: integrate dx/root 9x-4x^2 A. 1/9 sin^-1 (9x-8/8) + c…
Exercise 7.5
  1. x/(x+1) (x+2) Integrate the rational functions.
  2. 1/x^2 - 9 Integrate the rational functions.
  3. 3x-1/(x-1) (x-2) (x-3) Integrate the rational functions.
  4. x/(x-1) (x-2) (x-3) Integrate the rational functions.
  5. 2x/x^2 + 3x+2 Integrate the rational functions.
  6. 1-x^2/x (1-2x) Integrate the rational functions.
  7. x/(x^2 + 1) (x-1) Integrate the rational functions.
  8. x/(x-1)^2 (x+2) Integrate the rational functions.
  9. 3x+5/x^3 - x^2 - x+1 Integrate the rational functions.
  10. 2x-3/(x^2 - 1) (2x+3) Integrate the rational functions.
  11. 5x/(x+1) (x^2 - 4) Integrate the rational functions.
  12. x^3 + x+1/x^2 - 1 Integrate the rational functions.
  13. 2/(1-x) (1+x^2) Integrate the rational functions.
  14. 3x-1/(x+2)^2 Integrate the rational functions.
  15. 1/x^4 - 1 Integrate the rational functions.
  16. 1/x (x^n + 1) [Hint: multiply numerator and denominator by xn-1 and put xn=t]…
  17. cosx/(1-sinx) (2-sinx) [: x = t] Integrate the rational functions.…
  18. (x^2 + 1) (x^2 + 2)/(x^2 + 3) (x^2 + 4) Integrate the rational functions.…
  19. 2x/(x^2 + 1) (x^2 + 3) Integrate the rational functions.
  20. 1/x (x^4 - 1) Integrate the rational functions.
  21. 1/(6^x - 1) [Hint: Put ex = t] Integrate the rational functions.
  22. Choose the correct answer integrate xdx/(x-1) (x-2) A. log| (x-1)^2/x-2|+c B.…
  23. Integrate the rational functions.A. log|x| - 1/2 log (x^2 + 1) + c B. log|x| +…
Exercise 7.6
  1. xsinx Integrate the functions.
  2. xsin3x Integrate the functions.
  3. x^2e^x Integrate the functions.
  4. xlogx Integrate the functions.
  5. xlog2x Integrate the functions.
  6. x^2logx Integrate the functions.
  7. xsin^-1x Integrate the functions.
  8. xtan^-1x Integrate the functions.
  9. xcos^-1x Integrate the functions.
  10. (sin^-1x)^2 Integrate the functions.
  11. xcos^-1x/root 1-x^2 Integrate the functions.
  12. xsec^2x Integrate the functions.
  13. tan^-1x Integrate the functions.
  14. x (logx)^2 Integrate the functions.
  15. (x^2 + 1) logx Integrate the functions.
  16. e^x (sinx+cosx) Integrate the functions.
  17. xe^x/(1+x)^2 Integrate the functions.
  18. e^x (1+sinx/1+cosx) Integrate the functions.
  19. e^x (1/x - 1/x^2) Integrate the functions.
  20. (x-3) e^x/(x-1)^3 Integrate the functions.
  21. e^2xsinx Integrate the functions.
  22. sin^-1 (2x/1+x^2) Integrate the functions.
  23. integrate x^2e^x^3 dxequdls Choose the correct answer:A. 1/3 e^x^3 + c B. 1/3…
  24. integrate e^xsecx (1+tanx) dx Choose the correct answer:A. e^xcosx+c B.…
Exercise 7.7
  1. Integrate : root 4-x^2
  2. Integrate : root 1-4x^2
  3. Integrate : root x^2 + 4x+6
  4. Integrate : root x^2 + 4x+1
  5. Integrate : root 1-4x-x^2
  6. Integrate : root x^2 + 4x-5
  7. Integrate : root 1+3x-x^2
  8. Integrate : root x^2 + 3x
  9. Integrate : root 1 + x^2/9
  10. integrate root 1+x^2 dx is equal to Choose the correct answerA. x/2 root 1+x^2…
  11. integrate root x^2 - 8x+7 dx is equal to A. 1/2 (x-4) root x^2 - 2x+7 +…
Exercise 7.8
  1. integrate _a^bxdx Evaluate the following definite integrals as limit of sums.…
  2. integrate _0^5 (x+1) dx Evaluate the following definite integrals as limit of…
  3. integrate _2^3x^2 dx Evaluate the following definite integrals as limit of sums.…
  4. integrate _1^4 (x^2 - x) dx Evaluate the following definite integrals as limit…
  5. integrate _-1^1e^x dx Evaluate the following definite integrals as limit of…
  6. integrate _0^4 (x+e^2x) dx Evaluate the following definite integrals as limit of…
Exercise 7.9
  1. Evaluate integrate _-1^1 (x+1) dx
  2. Evaluate integrate _2^3 1/x dx
  3. Evaluate integrate _1^2 (4x^3 -5x^2 +6x+9) dx
  4. Evaluate integrate _0^pi /4 sin2xdx
  5. Evaluate integrate _0^pi /2 cos2xdx
  6. Evaluate integrate _4^5e^x dx
  7. Evaluate integrate _0^pi /4 tanxdx
  8. Evaluate integrate _ pi /6^pi /4 cosecxdx
  9. Evaluate integrate _0^1 dx/root 1-x^2
  10. Evaluate integrate _0^1 dx/1+x^2
  11. Evaluate integrate _2^3 dx/x^2 - 1
  12. Evaluate integrate _0^pi /2 cos^2xdx
  13. Evaluate integrate _2^3 xdx/x^2 + 1
  14. Evaluate integrate _0^1 2x+3/5x^2 + 1 dx
  15. Evaluate integrate _0^1xe^x^2 dx
  16. Evaluate integrate _1^2 5x^2/x^2 + 4x+3
  17. Evaluate integrate _0^pi /1 (2sec^2x+x^3 + 2) dx
  18. Evaluate integrate _0^pi (sin^2 x/2 - cos^2 x/2) dx
  19. Evaluate integrate _0^2 6x+3/x^2 + 4 dx
  20. Evaluate integrate _0^1 (xe^x + sin pi x/4) dx
  21. integrate _1^root 3 dx/x^2 + 1 equalsA. π/3 B. 2π/3 C. π/6 D. π/12…
  22. integrate dx/4+9x^2 equalsA. π/6 B. π/12 C. π/24 D. π/4

Exercise 7.1
Question 1.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x


Answer:

Method: To find the anti derivative of a function by inspection
Steps: 1. In this method we look for a function whose derivative is the given function. For Example: if we need to find anti derivative of x2, we know that derivative of x3 is 3x2. Therefore, the variable terms comes out to be the same.
2. After that balance out the coefficients of variables by dividing and multiplying suitable terms. From above example if [we divide x3 by 3 we will get the answer as x2. Hence, we can say that anti derivative of x2 is x3/3.

Now, similarly,

We know that


Therefore, the anti-derivative of sin2x is


Question 2.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

cos 3x


Answer:

We know that


Therefore, the anti-derivative of cos3x is .



Question 3.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

e2x


Answer:

We know that


Therefore, the anti-derivative of is .



Question 4.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

(ax + b)2


Answer:

We know that


Therefore, the anti-derivative of is .



Question 5.

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x – 4 e3x


Answer:

We know that



Therefore, the anti-derivative of sin2x is …(1)


Also,


Therefore, the anti-derivative of is …(2)


From (1) and (2), we get,



= sin 2x – 4e3x


Therefore, the anti-derivative of sin 2x – 4 e3x is .


Question 6.

Find the following integrals.



Answer:




Question 7.

Find the following integrals.



Answer:






Question 8.

Find the following integrals.



Answer:






Question 9.

Find the following integrals.



Answer:






Question 10.

Find the following integrals.



Answer:


Now we know that,

∫xn dx

Therefore,


Question 11.

Find the following integrals.



Answer:



Separating the terms we get,

Applying the formula,
∫xn dx =




Answer.

Question 12.

Find the following integrals.



Answer:


Separating the terms we get,

Applying the formula,

∫ xn dx =




Question 13.

Find the following integrals.



Answer:



Now the numerator can be factorized as,

x3 - x2 + x - 1 = x2(x - 1) + 1(x - 1)

x3 - x2 + x - 1 = (x2 + 1)(x - 1)

Now putting this in given integral we get,




Question 14.

Find the following integrals.



Answer:





Question 15.

Find the following integrals.



Answer:





Question 16.

Find the following integrals.



Answer:




Question 17.

Find the following integrals.



Answer:





Question 18.

Find the following integrals.



Answer:


Formulas Used: ∫ sec2x dx = tanx + c and ∫ secx tanx dx = sec x + c

Opening the brackets we get,


Answer.

Question 19.

Find the following integrals.



Answer:







= tanx –x +C



Question 20.

Find the following integrals.



Answer:




= 2tanx – 3secx + C



Question 21.

The anti-derivative of equals
A.

B.

C.

D.


Answer:



Question 22.

If such that f(2) = 0. Then f(x) is
A. B.

C. D.

Solution ||| The correct option is (A).


Answer:

It is given that



Also, It is given that f(2) = 0




Therefore,



Exercise 7.10
Question 1.

Evaluate the integrals using substitution.



Answer:

Given:

Let x2 + 1 = t


⇒ 2xdx = dt


⇒ xdx = � dt


When x = 0, t = 1 and when x = 1, t = 2








Question 2.

Evaluate the integrals using substitution.



Answer:

Given:

Let




Also, let


when,


so,








Question 3.

Evaluate the integrals using substitution.



Answer:

Given:

Let x = tan θ ⇒ dx = sec2 θ d θ


When, x = 0, θ = 0 and when x = 1, θ = π /4


Let






By applying product rule as,











Question 4.

Evaluate the integrals using substitution.



Answer:

Given:

Let x + 2 = t2 ⇒ dx = 2t dt


And x = t2 -2


when, x = 0, t = √2 and when x = 2, t = 2


so,













Question 5.

Evaluate the integrals using substitution.



Answer:

Given:

Let cos x = t


⇒ -sin xdx = dt


⇒ sin xdx = -dt


When x = 0, t = 1 and when x = π /2, t = 0



because,








Question 6.

Evaluate the integrals using substitution.



Answer:

Given:


we can write it as,




let


when



because,















Question 7.

Evaluate the integrals using substitution.



Answer:

Given:



Let x + 1 = t


⇒ dx = dt


When x = -1, t = 0 and when x = 1, t = 2



because,






Question 8.

Evaluate the integrals using substitution.



Answer:

Given:

Let 2x = t ⇒ 2 dx = dt


When x = 1, t = 2 and when x = 2, t = 4






now, let 1/t = f(t)


then, f' (t) = -1/t2



because,








Question 9.

The value of the integral is
A. 6

B. 0

C. 3

D. 4


Answer:

Given:

let


Now, let x = sin θ ⇒ dx = cos θ d θ


when, and when x = 1, θ = π/2









Now, let cot θ = t ⇒ - cosec2 θ d θ


when,









= 6


Correct option is: (A)


Question 10.

If then f'(x) is
A.

B.

C.

D.


Answer:

Given:

Applying product rule,






= -x cos x + sin x – 0


⇒ f(x) = -x cos x + sinx



⇒ f'(x) = -[{x(-sinx )} + cosx ] + cosx


= x sin x – cos x + cos x


= x sin x


Correct answer is B.



Exercise 7.11
Question 1.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:

let,


as,




Adding (1) and (2), we get









Question 2.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:

let,


as,




Adding (1) and (2), we get






Question 3.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:





Adding (1) and (2), we get









Question 4.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:





Adding (1) and (2), we get









Question 5.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:

As we can see that (x+2) ≤ 0 on [-5, -2] and (x+2) ≥ 0 on [-2,5]










Question 6.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:

As we can see that (x-5)≤0 on [2,5] and (x+2)≥0 on [5,8]









⇒ I = 9



Question 7.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:











Question 8.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:














Question 9.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:














Question 10.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:








Adding (1) and (2), we get











Question 11.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:


As we can see f(x) = sin2x and f(-x) = sin2(-x) = (sin (-x))2 = (-sin x)2 = sin2x.


i.e. f(x) = f(-x)


so, sin2x is an even function.


It is also known that if f(x) is an even function then,










Question 12.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:





Adding (1) and (2), we get










⇒ I = π



Question 13.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:


As we can see f(x) = sin7x and f(-x) = sin7(-x) = (sin (-x))7 = (-sin x)7 = -sin7x.


i.e. f(x) = -f(-x)


so, sin2x is an odd function.


It is also known that if f(x) is an odd function then,





Question 14.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:


As we see, f(x)=cos5x and f(2π –x) = cos5(2π –x) = cos5x = f(x)








Question 15.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:





Adding (1) and (2), we get





Question 16.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:





Adding (1) and (2), we get









Here, if f(x) = log (sin x) and f(π – x)=log ( sin (π –x))= log (sin x ) = f(x)





Adding (1) and (2), we get






Let 2x = t ⇒ 2dx = dt


When x = 0, t = 0 and when x = π /2, t = π







Question 17.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:




Adding (1) and (2), we get









Question 18.

By using the properties of definite integrals, evaluate the integrals



Answer:

Given:

As we can see that (x-1)≤0 when 0≤x≤1 and (x-1)≥0 when 1≤x≤4










Question 19.

Show that if f and g are defined as and


Answer:

Given:





Adding (1) and (2), we get








Question 20.

The value of
A. 0

B. 2

C.

D. 1


Answer:

Given:



It is also known that if f(x) is an even function then,



It is also known that if f(x) is an odd function then,






Correct answer is C


Question 21.

The value of
A. 2

B.

C. 0

D. -2


Answer:

Given:





Adding (1) and (2), we get







Correct answer is (c)



Miscellaneous Exercise
Question 1.

Integrate the function:


Answer:

Given:

Let


Using partial differentiation:


let




⇒ 1 = A – Ax2 + Bx – Bx2 + Cx + Cx2


⇒ 1 = A + (B+C)x + (-A-B+C)x2


Equating the coefficients of x, x2 and constant value. We get:


(a) A = 1


(b) B+C = 0 ⇒ B = -C


(c) -A –B +C =0


⇒ -1 – (-C) +C = 0


⇒ 2C = 1 ⇒ C = 1/2


So, B = -1/2


Put these values in equation (1)












Question 2.

Integrate the function:


Answer:

Given:

let


Multiply and divide by,












Question 3.

Integrate the function:


Answer:

Given:

let


put











Question 4.

Integrate the function:


Answer:

Given:

let


Multiply and divide by x-3, we get















Question 5.

Integrate the function:


Answer:

Given:

or we can write it as,


Let x = t6⇒ dx = 6t5dt




After division we get,









Question 6.

Integrate the function:


Answer:

Given:


Using partial differentiation:


let



⇒5x=A(x2+9)+(Bx+C)(x+1)


⇒ 5x = Ax2 +9A+ Bx2 +Bx+ Cx + C


⇒ 5x = 9A + C + (B+C)x + (A+B)x2


Equating the coefficients of x, x2 and constant value. We get:


(a) 9A + C = 0 ⇒ C = -9A


(b) B+C = 5 ⇒ B = 5-C ⇒ B = 5-(-9A) ⇒ B = 5 + 9A


( c) A + B =0 ⇒ A = -B ⇒ A = -(5 + 9A) ⇒ 10A = -5 ⇒ A = -1/2


and C = 9/2 and B = 1/2


Put these values in equation (1)










Put x2 = t ⇒ 2xdx = dt





Put the value in equ. (2)




Question 7.

Integrate the function:


Answer:

Given:


Let x – a = t ⇒ x = t + a ⇒ dx = dt



As,{ sin (A+B) = sin A cos B + cos A sin B}












Question 8.

Integrate the function:


Answer:

Given:










Question 9.

Integrate the function:


Answer:

Given:


Put sin x = t ⇒ cos x dx = dt







Question 10.

Integrate the function:


Answer:

Given:












Question 11.

Integrate the function:


Answer:

Given:


Multiply and divide by sin (a-b), we get





As,{ sin (A-B) = sin A cos B - cos A sin B}












Question 12.

Integrate the function:


Answer:

Given:


Now, let x4 = t ⇒ 4x3 dx = dt


And x3 dx = dt/4







Question 13.

Integrate the function:


Answer:

Given:


Let ex = t ⇒ ex dx = dt









Question 14.

Integrate the function:


Answer:

Given:


Using partial differentiation:




⇒1 = (Ax + B)(x2 + 4)+(Cx + D)(x2 + 1)


⇒ 1 = Ax3 +4Ax+ Bx2 + 4B+ Cx3 + Cx + Dx2 + D


⇒ 1 = (A+C)x3 +(B+D)x2 +(4A+C)x + (4B+D)


Equating the coefficients of x, x2, x3 and constant value. We get:


(a) A + C = 0 ⇒ C = -A


(b) B + D = 0 ⇒ B = -D


( c) 4A + C =0 ⇒ 4A = -C ⇒ 4A = A ⇒ 3A = 0 ⇒ A = 0 ⇒ C = 0


( d) 4B + D = 1 ⇒ 4B – B = 1 ⇒ B = 1/3 ⇒ D = -1/3


Put these values in equation (1)










Question 15.

Integrate the function:


Answer:

Given:

Let



Let cos x = t ⇒ -sin x dx = dt ⇒ sin x dx = dt








Question 16.

Integrate the function:


Answer:

Given:

Let




Let x4 = t ⇒ 4x3 dx = dt ⇒ x3 dx = dt/4








Question 17.

Integrate the function:


Answer:

Given: f (ax + b)[f(ax + b)]n

Let f(ax +b) = t ⇒ a .f (ax + b)dx = dt








Question 18.

Integrate the function:


Answer:

Given:


As,{ sin (A+B) = sin A cos B + cos A sin B}




















Question 19.

Integrate the function:


Answer:

Given:

let


As we know,








Now, first solve for I1:






because,












Put it in equ. (2)










Question 20.

Integrate the function:


Answer:

Given:

let


Let x= cos2θ ⇒ dx = -2sinθ cosθ dθ





















Question 21.

Integrate the function:


Answer:

Given:







Now let tan x = f(x)


⇒ f(x) = sec2x dx






Question 22.

Integrate the function:


Answer:

Given:

Let


Using partial differentiation:


let




⇒ x2 + x + 1 = Ax2 + 3Ax + 2A + Bx +2B + Cx2 + 2Cx + C


⇒ x2 + x + 1 = (2A+2B+C) + (3A+B+2C)x + (A+C)x2


Equating the coefficients of x, x2 and constant value. We get:


(a) A + C = 1


(b) 3A + B + 2C = 1


( c) 2A+2B+C =1


After solving we get:


A=-2, B=1 and C=3









Question 23.

Integrate the function:


Answer:

Given:


Let x= cosθ ⇒ dx = -sin θ d θ
















Question 24.

Integrate:


Answer:

Given:


















Question 25.

Evaluate the definite integral:


Answer:

Given:















Question 26.

Evaluate the definite integral:


Answer:

Given:




Now let tan2x = t ⇒ 2 tan x sec2x dx = dt


And when x=0 then t=0 and when x=π /4 then t=1








Question 27.

Evaluate the definite integral:


Answer:

Given:














First solve for I1:



Let 2 tan x = t ⇒ 2 sec2x dx dt


When x = 0 then t = 0 and when x = π /2 then t = ∞







Put this value in equ.(2)





Question 28.

Evaluate the definite integral:


Answer:

Given:






Now let sin x – cos x = t ⇒( cos x + sin x )dx = dt






i.e. f(x) = f(-x)


so, f(x) is an even function.


It is also known that if f(x) is an even function then,







Question 29.

Evaluate the definite integral:


Answer:

Given:















Question 30.

Evaluate the definite integral:


Answer:

Let


Also, let sinx – cosx = t


Differentiating both sides, we get,


(cosx + sinx) dx = dt


When x = 0, t = -1


And when , t = 0


Now,


(sinx – cosx)2 = t2


1 – 2 sinx.cosx = t2


Sin2x = 1 – t2


Putting all the values, we get the integral,









Question 31.

Evaluate the definite integral:


Answer:

Given:



Let sin x = t ⇒ cos x dx = dt


When x =0 then t = 0 and when x = π /2 then t = 1

















Question 32.

Evaluate the definite integral:


Answer:

Given:







Adding (1) and (2), we get



















Question 33.

Evaluate the definite integral:


Answer:

Given:




First solve for I1:



As we can see that (x-1)≥0 when 1≤x≤4








Now solve for I2:



As we can see that (x-2)≤0 when 1≤x≤2 and (x-2)≥0 when 2≤x≤4









Now solve for I3:



As we can see that (x-3)≤0 when 1≤x≤3 and (x-3)≥0 when 3≤x≤4













Question 34.

Prove:


Answer:

Given:



Using partial differentiation:





⇒ 1 = Ax2 +Ax+ B+Bx+ Cx2


⇒ 1 = B + (A+B)x + (A+C)x2


Equating the coefficients of x, x2 and constant value. We get:


(a) B = 1


(b) A + B = 0 ⇒ A = -B ⇒ A = -1


( c) A + C =0 ⇒ C = -A ⇒ C = 1


Put these values in equation (1)











⇒ L.H.S = R.H.S


Hence proved.



Question 35.

Prove:


Answer:

Given:










L.H.S = R.H.S


Hence Proved.



Question 36.

Prove:


Answer:

Given:



As we can see f(x) =x17 .cos4x and f(-x) = (-x)17 .cos4(-x) = -x17 .cos4x


i.e. f(x) = -f(-x)


so, it is an odd function.


It is also known that if f(x) is an odd function then,




Hence proved.



Question 37.

Prove:


Answer:

Given:







First solve for I1:



Let cos x = t ⇒ -sin x dx = dt ⇒ sinx dx = -dt


When x=0 then t= 1 and when x = π /2 then t = 0







Put in equ. (2)






L.H.S = R.H.S


Hence Proved.



Question 38.

Prove:


Answer:

Given:







First solve for I1:



Let tan x = t ⇒ sec2 x dx = dt


When x=0 then t= 0 and when x = π /2 then t = 1





Put in equ. (2)








L.H.S = R.H.S


Hence Proved.



Question 39.

Prove:


Answer:

Given:







First solve for I1:



Let 1 - x2 = t ⇒ -2 x dx = dt


When x=0 then t= 1 and when x = 1 then t = 0






Put in equ. (2)





L.H.S = R.H.S


Hence Proved.



Question 40.

Evaluate as a limit of a sum.


Answer:

Given:




Here, a=0, b=1, and f(x)=e2-3x and h = 1/n















Question 41.

Choose the correct answers

is equal to
A.

B.

C.

D.


Answer:

Given:




Put ex= t ⇒ ex dx = dt





Hence, correct option is (A).


Question 42.

Choose the correct answers

is equal to
A.

B.

C.

D.


Answer:

Given:





Put sin x + cos x= t ⇒ cos x – sin x = dt





Hence, correct option is (B).


Question 43.

Choose the correct answers

If f (a + b – x) = f (x), then is equal to
A.

B.

C.

D.


Answer:

Given:









Hence, correct option is (D).


Question 44.

Choose the correct answers

The value of
A. 1

B. 0

C. –1

D.


Answer:

Given:









Adding (1) and (2), we get






Hence, correct option is (B).



Exercise 7.2
Question 1.

Integrate the functions.



Answer:

Let 1+x2 = t

⇒ 2x dx = dt


Now,


= log|t| + C


= log|1+x2| + C


= log(1+x2) + C


Question 2.

Integrate the functions.



Answer:

Let log|x| = t


Now,





Question 3.

Integrate the functions.



Answer:


let 1+logx =t




= log|t| + C


= log | 1+ logx| + C



Question 4.

Integrate the functions.

sin x sin (cos x)


Answer:

Let cosx = t

⇒ -sinxdx = dt



=-[-cost] + C


= cost + C


= cos(cosx) + C



Question 5.

Integrate the functions.
sin (ax + b) cos (ax + b)


Answer:

Let I = ∫sin (ax + b) cos (ax + b) dx

We know that,

sin 2A = 2sinA.cosA

Therefore,

sin (ax + b) cos (ax + b) =

Let 2(ax+b) = t


⇒ 2adx = dt



=



Question 6.

Integrate the functions.



Answer:

Let ax + b =t

⇒ adx = dt





Question 7.

Integrate the functions.



Answer:

Let (x +2) = t

⇒ dx = dt






Question 8.

Integrate the functions.



Answer:

Let 1 +2x2 =t

⇒ 4xdx = dt







Question 9.

Integrate the functions.



Answer:

Let x2 + x+ 1= t

Differentiating both sides, we get,

⇒ (2x + 1)dx = t dt

Therefore,






Question 10.

Integrate the functions.



Answer:


Now, Let




= 2log|t| + C


= 2log|| + C



Question 11.

Integrate the functions.



Answer:

Let x+ 4 = t

⇒ dx = dt











Question 12.

Integrate the functions.



Answer:

Let x3 -1 = t

⇒ 3x2dx = dt









Question 13.

Integrate the functions.



Answer:

Let 2 +3x3 = 1

⇒ 9x2 dx = dt







Question 14.

Integrate the functions.



Answer:

Let log x = t






Question 15.

Integrate the functions.



Answer:

Let 9 – 4x2 = t

⇒ -8xdx = dt






Question 16.

Integrate the functions.



Answer:

Let 2x +3 =t

⇒ 2dx = dt






Question 17.

Integrate the functions.



Answer:

Let x2 = t

⇒ 2xdx = dt








Question 18.

Integrate the functions.



Answer:

let tan-1 x = t



= et + C


=



Question 19.

Integrate the functions.



Answer:

We have,

Dividing numerator and denominator by ex, we get,




Let ex + e-x = t
Differentiating both sides, we get,

(ex - e-x )dx = dt

Now the integral becomes,

= log |t| +C


= log|ex + e-x| + C


Question 20.

Integrate the functions.



Answer:

Let






= log |t| +C


= log|| + C



Question 21.

Integrate the functions.

tan2 (2x – 3)


Answer:

tan2 (2x – 3) = sec2 (2x – 3) – 1

Let 2x -3 = t


⇒ 2dx = dt






⇒ 1/2 tan t - t + C

⇒ 1/2 tan (2x - 3) - (2x - 3) + C


Question 22.

Integrate the functions.

sec2 (7 – 4x)


Answer:

Let 7 – 4x = t

⇒ -4dx = dt






Question 23.

Integrate the functions.



Answer:

let sin-1 x =t






Question 24.

Integrate the functions.



Answer:


let 3cosx + 2sinx = t


(-3sinx + 2cosx)dx = dt







Question 25.

Integrate the functions.



Answer:


Let (1 – tanx) = t


⇒ -sec2xdx = dt






Question 26.

Integrate the functions.



Answer:

Let



= 2sint + C


= 2sin + C



Question 27.

Integrate the functions.



Answer:

Let sin2x = t

⇒ 2cos2xdx = dt


.





Question 28.

Integrate the functions.



Answer:

Let 1 + sinx = t

⇒ cosx dx = dt







Question 29.

Integrate the functions.

cot x log sin x


Answer:

Let Log sinx = t


⇒ cotx dx = dt






Question 30.

Integrate the functions.



Answer:

Let 1+ cosx = t

⇒ -sinx dx = dt



= - log|t| + C


= -log|1 + cosx| + C



Question 31.

Integrate the functions.



Answer:

Let 1 + cosx = t

⇒ -sinx dx = dt







Question 32.

Integrate the functions.



Answer:

Let I =







Let sinx + cosx = t


⇒ (cosx-sinx)dx = dt


Therefore, I =





Question 33.

Integrate the functions.



Answer:

Let I =








Let cosx - sinx = t


⇒ (-sinx-cosx)dx = dt


Therefore, I =





Question 34.

Integrate the functions.



Answer:

Let I =




Let tanx = t


⇒ sec2x dx = dt


⇒ I =


= 2+C


= 2 + C



Question 35.

Integrate the functions.



Answer:

let 1 + log x = t

= dt






Question 36.

Integrate the functions.



Answer:


Let (x+logx) = t







Question 37.

Integrate the functions.



Answer:

Let x4 = t

⇒ 4 x3 dx = dt




Let tan-1 = v



Thus, we get,







Question 38.

Choose the correct answer:


A. 10x – x10 + C
B. 10x + x10 + C
C. (10x – x10)–1 + C
D. log (10x + x10) + C


Answer:

Let x10 + 10x = t

⇒ (10x9 + 10x loge10)dx = dt



= log t + C


= log(x10 + 10x) + C


Question 39.

equals
A. tan x + cot x + C B. tan x – cot x + C

C. tan x cot x + C D. tan x – cot 2x + C


Answer:

Let I =




= tanx – cotx + C



Exercise 7.3
Question 1.

Find the integrals of the functions.



Answer:

sin2(2x+5)







Question 2.

Find the integrals of the functions.



Answer:








Question 3.

Find the integrals of the functions.



Answer:








Question 4.

Find the integrals of the functions.



Answer:

Let I = sin3(2x+1)



Let cos (2x+1) = t


=> -2sin(2x+1)dx = dt


=> sin(2x+1)dx =







Question 5.

Find the integrals of the functions.



Answer:

Let I =



Let cosx = t


⇒ -sinx.dx = dt


⇒ I







Question 6.

Find the integrals of the functions.



Answer:











Question 7.

Find the integrals of the functions.



Answer:






Question 8.

Find the integrals of the functions.


Answer:






Question 9.

Find the integrals of the functions.



Answer:








Question 10.

Find the integrals of the functions.



Answer:

sin4x = sin2xsin2x







Now,






Question 11.

Find the integrals of the functions.



Answer:

cos42x = (cos22)2







Now,




Question 12.

Find the integrals of the functions.



Answer:




= 1- cosx



= x – sinx + C



Question 13.

Find the integrals of the functions.



Answer:

Using the identity , we have



Now, using the identity sin 2x = 2 sin x cos x, we have



Using identity 2 cos A cos B = cos (A + B) + cos (A - B), we have


= 2[cos(x) +cosα]


= 2cosx + 2 cosα



= 2[sinx + xcosα] + C


Question 14.

Find the integrals of the functions.



Answer:




Let sinx + cosx = t


⇒ (cosx-sinx)dx =dt




= -t-1 + C





Question 15.

Find the integrals of the functions.



Answer:

tan32xsec2x = tan22xtan2xsec2x

= (sec22x -1)tan2xsec2x


= sec22x.tan2xsec2x-tan2xsec2x




Now, Let sec2x = t


⇒ 2sec2xtan2x dx = dt


Thus,





Question 16.

Find the integrals of the functions.



Answer:

tan4x = tan2x.tan2x

= (sec2x-1) tan2x


= sec2x tan2x- tan2x


= sec2x tan2x- (sec2x-1)


= sec2x tan2x- sec2x+1


Now,



Now, let tanx = t


=> sec2x dx =dt





Question 17.

Find the integrals of the functions.



Answer:



= tanxsecx + cotxcosecx


⇒ Now,


= secx – cosecx + C



Question 18.

Find the integrals of the functions.



Answer:



= sec2x


Now,


= tanx + C



Question 19.

Find the integrals of the functions.



Answer:




Now,


let tanx = t


⇒ sec2x dx = dt






Question 20.

Find the integrals of the functions.



Answer:


Now,


Let 1 + sin2x = t


⇒ 2cos2x dx = dt


Thus,





= log|sinx + cosx| + C



Question 21.

Find the integrals of the functions.



Answer:

sin-1(cosx)

Let cosx = t.......(1)

Then, sinx =

Differentiating both sides of (1), we get,

⇒ (-sinx)dx = dt


⇒ dx =


⇒ dx =


⇒ Now,



Let sin-1 t = v



⇒ ∫sin-1(cosx) dx = - ∫vdv




…(1)


We know that,


sin-1x + cos-1 x =


⇒ sin-1(cosx)=


Now, substituting in eq(1), we get,







Question 22.

Find the integrals of the functions.



Answer:





Now,





Question 23.



A. B.

C. D.


Answer:


We know that,
∫sec2x dx = tan x + c
∫cosec2x dx = - cot x + c
= tanx + cotx + C.
A: Answer


Question 24.


A. B.

C. D.


Answer:

Let x.ex = t
Differentiating both sides we get,

⇒ (ex.x + ex.1)dx = dt


⇒ ex(x + 1) = dt


Now,



= tan t + C


= tan(ex.x) + C



Exercise 7.4
Question 1.

Integrate the functions.



Answer:

Let x3 = t

⇒ 3x2 dx = dt



= tan-1t + C


= tan-1(x3) + C



Question 2.

Integrate the functions.



Answer:

Let 2x= t

⇒ 2dx = dt





Question 3.

Integrate the functions.



Answer:

Let 2 – x = t

⇒ -dx = dt







Question 4.

Integrate the functions.



Answer:

Let 5x = t

⇒ 5dx = dt






Question 5.

Integrate the functions.



Answer:

Let = t

⇒ 2dx = dt






Question 6.

Integrate the functions.



Answer:

Let x3 = t

⇒ 3x2 dx = dt






Question 7.

Integrate the functions.



Answer:


For ,


Let x2 -1 = t


⇒ 2x dx = dt








Question 8.

Integrate the functions.



Answer:

Let x3 = t

⇒ 3x2 dx = dt






Question 9.

Integrate the functions.



Answer:

Let tanx = t

⇒ sec2xdx = dt






Question 10.

Integrate the functions.



Answer:


Let x+1 = t


⇒ dx = dt







Question 11.

Integrate the functions.



Answer:


Let 3x+1= t


⇒ 3dx = dt






Question 12.

Integrate the functions.



Answer:


Let x+3= t


⇒ dx = dt






Question 13.

Integrate the functions.



Answer:


Let


⇒ dx = dt






Question 14.

Integrate the functions.



Answer:


Let


⇒ dx = dt






Question 15.

Integrate the functions.



Answer:

(x - 1)(x - b) can be written as x2 – (a+b)x + ab.

Then, x2 – (a+b)x + ab = x2 – (a+b)x +




Let x =


⇒ dx = dt






Question 16.

Integrate the functions.



Answer:

Let 4x + 1 =

⇒ 4x + 1 = A(4x + 1) + B


⇒ 4x + 1 = 4Ax+ A+ B


Now, equating the coefficients of x and constant term on both sides, we get,


4A = 4


⇒ A = 1


A + B = 1


⇒ B = 0


Let 2x2 + x – 3 =t


⇒ (4x + 1) dx = dt




2



Question 17.

Integrate the functions.



Answer:

Let x + 2 =

⇒ x + 2 = A(2x) + B


Now, equating the coefficients of x and constant term on both sides, we get,


2A = 1


⇒ A =


B = 2


⇒ x + 2 = (2x) + 2




Now,


Let x2 - 1 = t


⇒ (2x)dx = dt




And




Question 18.

Integrate the functions.



Answer:

Let 5x - 2 =

⇒ 5x - 2 = A(2 + 6x) + B


Now, equating the coefficients of x and constant term on both sides, we get,


6A = 5


⇒ A =


2A + B = -2


⇒ B =


⇒ 5x - 2 = (2 + 6x) +




Now,


Let 1+2x+3x2 = t


⇒ (2 + 6x)dx = dt



= log|1+2x+3x2| …(1)


And


1+2x+3x2 =






…(2)


Thus, from (1) and (2), we get,





Question 19.

Integrate the functions.



Answer:

Let 6x + 7 =

⇒ 6x + 7 = A(2x - 9) + B


Now, equating the coefficients of x and constant term on both sides, we get,


2A = 6


⇒ A = 3


-9A + B = 7


⇒ B = 34


⇒ 6x + 7 = 3 (2x - 9) + 34




Now,


Let x2 – 9x + 20 = t


⇒ (2x - 9)dx = dt


----------(1)


And


x2 – 9x + 20 =




…(2)


Thus, from (1) and (2), we get,





Question 20.

Integrate the functions.



Answer:

Let x + 2 =

⇒ x + 2 = A(4 -2x) + B


Now, equating the coefficients of x and constant term on both sides, we get,


-2A = 1


⇒ A =


4A + B = 2


⇒ B = 4


⇒ x + 2 =


Now,



Now, let us consider,


Let 4x – x2 = t


⇒ (4 -2x) dx= dt


…(1)


And, Now let us consider,


Then, 4x – x2 = -(-4x + x2)


= (-4x + x2 + 4 – 4)


= 4 – (x - 2)2


= (2)2 – (x - 2)2


…(2)


using eq. (1) and (2), we get,




Question 21.

Integrate the functions.



Answer:





Now, Let us consider


Let x2 + 2x + 3 = t


⇒ (2x + 2)dx = dt


… (1)


And, now let us consider


⇒ x2 + 2x + 3 = x2 + 2x + 1 + 2 = (x +1)2 + (2



Using eq. (1) and (2), we get,





Question 22.

Integrate the functions.



Answer:

Let x + 3 =

⇒ x + 3 = A(2x-2) + B


Now, equating the coefficients of x and constant term on both sides, we get,


2A = 1


⇒ A =


-2A + B = 3


⇒ B = 4


⇒ x + 3 =


Now,



Now, Let us consider


Let x2 – 2x – 5 = t


⇒ (2x -2)dx = dt



And, now let us consider,




…(2)


Using eq. (1) and (2), we get,





Question 23.

Integrate the functions.



Answer:

Let 5x + 3 =

⇒ 5x + 3 = A(2x + 4) + B


Now, equating the coefficients of x and constant term on both sides, we get,


2A = 5


⇒ A =


4A + B = 3


⇒ B = -7


⇒ 5x + 3 =


Now,



Now, let us consider,


Let x2 +4x + 10= t


⇒ (2x + 4) dx= dt


… (1)


And, Now let us consider,




…(2)


using eq. (1) and (2), we get,





Question 24.

Choose the correct answer:


A.

B.

C.

D.


Answer:




Question 25.

Choose the correct answer:


A.

B.

C.

D.


Answer:







Exercise 7.5
Question 1.

Integrate the rational functions.



Answer:

Let

⇒ x = A(x + 2) + B(x +1)


On comparing the coefficients of x and constant term, we get,


A + B = 1


2A + B = 0


On solving above two equations, we get,


A = -1 and B = 2


Thus,




= -log|X + 1| + 2log|x + 2| + C


= log (x+2)2 – log |x + 1| + C




Question 2.

Integrate the rational functions.



Answer:


Let


⇒ 1= A(x - 3) + B(x + 3)


On comparing the coefficients of x and constant term, we get,


A + B = 0


-3A + 3B = 1


On solving above two equations, we get,


A = and B =


Thus,







Question 3.

Integrate the rational functions.



Answer:

Let

⇒ 3x -1 = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2) …(1)


Substituting x = 1, 2 and 3 respectively in equation (1), we get,


A = 1, B = -5 and C = 4


Thus,




= log|x -1| -5log|x-2| + 4log|x-3| + C



Question 4.

Integrate the rational functions.



Answer:

Let

⇒ x = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2) …(1)


Substituting x = 1, 2 and 3 respectively in equation (1), we get,


A = , B = -2 and C =


Thus,






Question 5.

Integrate the rational functions.



Answer:

Let

⇒ 2x = A(x + 2)+ B(x + 1) …(1)


Substituting x = -1 and -2 respectively in equation (1), we get,


A = -2, B = 4


Thus,




= 4log|x + 2| -2log|x + 1| + C



Question 6.

Integrate the rational functions.



Answer:

On dividing 1 – x2 by x(1 - 2x), we get,


…(1)


Now, let


(2 – x) = A(1 – 2x) + Bx …(2)


Now, substituting x = 0 and in equation (2), we get,


A = 2 and B = 3


Thus,


Now, putting this value in equation (2), we get,






Question 7.

Integrate the rational functions.



Answer:

Let

x = (Ax + B)(x -1) + C (x2 + 1)


⇒ x = Ax2 – Ax + Bx – B + Cx2 + C


Equating the coefficients of x2, x and constant term, we get,


A + C = 0


-A + B = 0


-B + c = 0


On solving these equation, we get,


A = , B = and C =


Thus,





Now, let us consider, ,


Let (x2 + 1) = t


2xdx = dt


Thus,



Therefore,





Question 8.

Integrate the rational functions.



Answer:

Let

⇒ x = A(x - 1)(x + 2) + B(x + 2) + C( x – 1)2 …(1)


Substituting x = -1 in equation (1), we get,


B =


Equating the coefficients of x2 and constant term, we get,


A + C = 0


-2A + 2B + C = 0


A = and c =


Thus,








Question 9.

Integrate the rational functions.



Answer:


Let


⇒3x + 5 = A(x - 1)(x + 1) + B(x + 1) + C( x – 1)2


⇒3x + 5 = A(x2 - 1) + B(x + 1) + C( x2 + 1 – 2x) …(1)


Substituting x = 1 in equation (1), we get,


B = 4


Equating the coefficients of x2 and x, we get,


A + C = 0


b – 2C = 3


A = and c =


Thus,








Question 10.

Integrate the rational functions.



Answer:


Let


⇒2x – 3 = A(x - 1)(2x + 3) + B(x + 1)(2x + 3) + C(x+1)( x – 1)


⇒ 2x – 3 = A(2x2 + x – 3) + B(2x2 + 5x + 3) + C(x2 - 1)


⇒2x - 3 = (2A + 2B + C)x2 + (A + 5B)x + (-3A + 3B – C)


Equating the coefficients of x2 and x, we get,


B =, A = and C =


Thus,








Question 11.

Integrate the rational functions.



Answer:


Let


⇒ 5x = A(x + 2)(x - 2) + B(x + 1)(x - 2) + C(x+1)( x + 2) …(1)


Substituting x = -1, -2 and 2 respectively in equation (1), we get,


A =, B =, and C =


Thus,







Question 12.

Integrate the rational functions.



Answer:


Dividing (x3 + x + 1) by x2 -1, we get,



Let


Now, 2x + 1 = A(x – 1) + B(x + 1) …(1)


Substituting x = 1 and -1 in equation (1), we get,


A = and B =


Thus,





Question 13.

Integrate the rational functions.



Answer:

Let

⇒ 2 = A(1 – x2) + (Bx + C)(1 – x)


⇒ 2 = A + Ax2 + Bx – Bx2 + C – Cx


On comparing the coefficients of x2, x and constant term, we get,


A – B = 0


B – C = 0


A + C = 0


On solving these equations, we get,


A = 1, B =1 and C = 1


Thus,







Question 14.

Integrate the rational functions.



Answer:

Let

⇒ 3x -1 = A(x + 2) + B


Equating the coefficients of x and constant term, we get,


A = 3


2A + B = -1


B = -7


Thus,







Question 15.

Integrate the rational functions.



Answer:


Let


1 = A(x-1)(x2+1) + B(x+1)(x2+1) + (Cx + D)(x2 - 1)


1 = A(x3 + x – x2 -1) + B(x3 + x + x2 + 1) + Cx3 + Dx2 - Cx - D


1 = (A + B + C)x3 + (-A + B + D)x2 + (A + B - C)x + (-A + B - D)


Equating the coefficients of x3, x2, x and constant term, we get,


(A + B + C) = 0


(-A + B + C) = 0


(A + B - C) = 0


(-A + B - D) = 0


On solving these equations, we get,


A =


Therefore,







Question 16.

Integrate the rational functions.

[Hint: multiply numerator and denominator by xn-1 and put xn=t]


Answer:


Multiplying numerator and denominator by xn-1, we get,



Let xn = t


xn-1dx = dt


Therefore,



Let


1 = A(1+t) + Bt ...(1)


Substituting t = 0, -1 in equation (1), we get,


A =1 and B = -1


Thus,








Question 17.

Integrate the rational functions.



Answer:


Let sinx = t


cosx dx = dt


Therefore,



Let


1 = A(2 – t) + B (1 – t) …(1)


Substituting t = 2 and then t = 1 in equation (1), we get,


Therefore,








Question 18.

Integrate the rational functions.



Answer:

Now,

Let


4x2 + 10 = (Ax + B)(x2 + 4) + (Cx + D)(x2 + 3)


⇒ 4x2 + 10 = Ax3 + 4Ax + Bx2 + 4B + Cx3 + 3Cx + Dx2 + 3D


⇒ 4x2 + 10 = (A+C)x3 + (B + D)x2 + (4A + 3C)x + (4B + 3D)


Equating the coefficients of x3, x2, x and constant term, we get,


A + C = 0


B + D = 0


4A + 3C = 0


4B + 3 B = 0


On solving these equations, we get,


A = 0, B = -2, C = 0 and D = 6


Therefore,









Question 19.

Integrate the rational functions.



Answer:

Now,

Now, let x2 = t


2xdx = dt


Thus, …(1)


Let


1 = A(t + 3) + B(t + 1)……………….(2)


Substituting t = -3 and -1 in (2), we get


A = and B =








Question 20.

Integrate the rational functions.



Answer:

Now,

Multiplying numerator and denominator by x3, we get,




Now, let x4 = t


4x3dx = dt


Thus,


Let


1 = A(t – 1) + Bt …(1)


Substituting t = 0 and 1 in (1), we get


A = -1 and B = 1








Question 21.

Integrate the rational functions.

[Hint: Put ex = t]


Answer:

Now,

Let ex = t


ex dx = dt



Let


1 = A(t – 1) + Bt …(1)


Substituting t =1 and t = 0 in equation (1), we get,


A = -1 and B = 1






Question 22.

Choose the correct answer


A.
B.

C.

D.


Answer:

Let

x = A(x – 2) + B(x - 1) …(1)


Substituting x =1 and 2 in (1), we get,


A = -1 and B = 2




= -log|x – 1| + 2log| x – 2| + C



Question 23.

Integrate the rational functions.


A.

B.

C.

D.


Answer:

Let

1 = A(X2 + 1) + (Bx + C)


Equating the coefficients of x2, x and constant term, we get,


A + B = 0


C = 0


A = 1


On solving these equations, we get,


A = 1, B = -1 and C = 0






Exercise 7.6
Question 1.

Integrate the functions.



Answer:

Let I = xsinx

Now, integrating by parts, we get,






Question 2.

Integrate the functions.



Answer:

Let I = sin3x

Now, integrating by parts, we get,






Question 3.

Integrate the functions.



Answer:

Let I = x2ex

Now, integrating by parts, we get,





Again integrating by parts, we get,








Question 4.

Integrate the functions.



Answer:

Let I = xlogx

Now, integrating by parts, we get,

Taking, Logarithmic function as first function and algebraic function as second function,



Question 5.

Integrate the functions.



Answer:

Let I = xlog2x

Now, integrating by parts, we get,







Question 6.

Integrate the functions.



Answer:

Let I = x2logx

Now, integrating by parts, we get,







Question 7.

Integrate the functions.



Answer:

Let I = xsin-1x

Now, integrating by parts, we get,











Question 8.

Integrate the functions.



Answer:

Let I = xtan-1x

Now, integrating by parts, we get,










Question 9.

Integrate the functions.



Answer:

Let I = xcos-1x

Now, integrating by parts, we get,







…(1)


Now,









Now, substituting in (1), we get,





Question 10.

Integrate the functions.



Answer:

Let I = (sin-1x)2

Now, integrating by parts, we get,










Question 11.

Integrate the functions.



Answer:

Let


Now, integrating by parts, we get,








Question 12.

Integrate the functions.



Answer:

Let I = xsec2x

Now, integrating by parts, we get,




= xtanx + log|cosx| + C



Question 13.

Integrate the functions.



Answer:

Let

So, now integrating by parts, we get,








Question 14.

Integrate the functions.



Answer:

Let

Integrating by parts, we get,







Question 15.

Integrate the functions.



Answer:


Now, Let I = I1 + I2 …(1)


Where,


So, now



Integrating by parts, we get,





…(2)


Now,


Integrating by parts, we get,





= xlogx –x + C2 …(3)


Now putting the value of I1 and I2, in (1), we get,


I





Question 16.

Integrate the functions.



Answer:


Now,


Let


We know that,



Thus,




Question 17.

Integrate the functions.



Answer:




Now,


Let


We know that,



Thus,




Question 18.

Integrate the functions.



Answer:









Now,


Let


We know that,



Thus,




Question 19.

Integrate the functions.



Answer:

Let I

Now, let



We know that,



Thus,




Question 20.

Integrate the functions.



Answer:



Now, let f(x) =



We know that,



Thus,




Question 21.

Integrate the functions.



Answer:

Let I = e2xsinx

Integrating by parts, we get,


I




Again, integrating by parts, we get,










Question 22.

Integrate the functions.



Answer:

Let x = tanƟ

⇒ dx = sec2ƟdƟ


Thus,




Integrating by parts, we get




= 2[ƟtanƟ + log|cosƟ| + C







Question 23.

Choose the correct answer:


A.

B.

C.

D.


Answer:

Let I =

Also, let x3 = t


⇒ 3x2dx = dt


Thus,


⇒ I =




Question 24.

Choose the correct answer:

A.

B.

C.

D.


Answer:


Let I =


Also, let secx = f(x)


⇒ secxtanx = f’(x)


We know that,


Thus, I = exsecx + C



Exercise 7.7
Question 1.

Integrate :



Answer:


We know that,



Therefore,





Question 2.

Integrate :


Answer:


Let 2= t




We know that,



Therefore,







Question 3.

Integrate :


Answer:




⇒ We know that,



Therefore,





Question 4.

Integrate :


Answer:




We know that,



Therefore,




Question 5.

Integrate :


Answer:





We know that,





Question 6.

Integrate :


Answer:




We know that,





Question 7.

Integrate :


Answer:





We know that,



Therefore,





Question 8.

Integrate :


Answer:




We know that,



Therefore,





Question 9.

Integrate :


Answer:




We know that,



Therefore,





Question 10.

Choose the correct answer

is equal to
A.

B.

C.

D.


Answer:

We know that,



Therefore,



Question 11.

Choose the correct answer

is equal to

A.

B.

C.

D.


Answer:




We know that,



Therefore,




Exercise 7.8
Question 1.

Evaluate using limit of sums


Answer:

f(x) is continuous in [a,b]


here h=b-a/n











Question 2.

Evaluate using limit of sums


Answer:

f(x) is continuous in [0,5]


here h=5/n










Question 3.

Evaluate using limit of sums


Answer:

f(x) is continuous in [2,3]


here h=1/n














Question 4.

Evaluate using limit of sums


Answer:

f(x) is continuous in [1,4]


here h=3/n














Question 5.

Evaluate using limit of sums


Answer:

f(x) is continuous in [-1,1]


here h=2/n







Which is g.p with common ratio e1/n.


Whose sum is





=-1



As h=2/n




=e-e-1


Question 6.

Evaluate using limit of sums


Answer:


F(x)=h(x)+g(x)


Solving for h(x)


. h(x) is continuous in [0,4]



here h=4/n









=8


Now solving for g(x)


g(x) is continuous in [0,4]



here h=4/n







Which is g.p with common ratio e1/n


Whose sum is






As h=4/n



=(e8-1)


Now for f(x)=h(x)+g(x)


=8+ e8-1



Exercise 7.9
Question 1.

Evaluate


Answer:

Let I =






⇒ I = 2




Question 2.

Evaluate


Answer:

Let I =

[]



⇒ I = log |3| - log |2|


⇒ I = log 3/2




Question 3.

Evaluate


Answer:

Let I =




[]



Question 4.

Evaluate


Answer:

Let I =

⇒ I =


[∫sinx dx=-cosx]


⇒ I = - (cos 2×π/4 – cos 0)/2


⇒ I = - (cos π/2 – cos 0)/2 = - (0 – 1)/2


⇒ I = 1/2


= 1/2


Question 5.

Evaluate


Answer:

Let I =

⇒ I =


[]




⇒ I = � × (0 – 0) = 0


= 0



Question 6.

Evaluate


Answer:

Let I =

⇒ I =


⇒ I = = e5 – e4[]


⇒ I = e4 (e – 1)


= e4 (e – 1)



Question 7.

Evaluate


Answer:

Let I =

⇒ I = []


⇒ I = =


⇒ I =


⇒ I =


=



Question 8.

Evaluate


Answer:

Let I =

⇒ I =


[]


⇒ I = log |cosec π/4 – cot π/4| - log |cosec π/6 – cot π/6|


⇒ I = log |√2 – 1| - log |2 - √3|



=



Question 9.

Evaluate


Answer:

Let I =

⇒ I = []


⇒ I =


⇒ I = sin-1(1) – sin-1(0) = π/2 – 0


⇒ I = π/2


= π/2



Question 10.

Evaluate


Answer:

Let I =

⇒ I =

We know that,

Therefore,

⇒ I =


⇒ I = tan-1(1) – tan-1(0) = π/4 – 0


⇒ I = π/4


= π/4


Question 11.

Evaluate


Answer:

Let I =

⇒ I = []


⇒ I = =


⇒ I = =


⇒ I = � log 3/2


= � log 3/2



Question 12.

Evaluate


Answer:

Let I =

cos 2x = 2cos2x – 1


cos2x =


putting the value cos2x in I


⇒ I = []


⇒ I =


⇒ I = = π/4


= π/4



Question 13.

Evaluate


Answer:

Let I =

Let x2 + 1 = t …(i)


∴ d(x2 + 1) = dt


⇒ 2x dx = dt


⇒ x dx = dt/2


When x = 2; t = 22 + 1 = 5


When x = 3; t = 32 + 1 = 10


Substituting x2 + 1 and x dx in I


⇒ I = []


⇒ I =


⇒ I = � log 2


= � log 2



Question 14.

Evaluate


Answer:

Let I =

Multiplying by 5 in numerator and denominator


⇒ I =


⇒ I =


⇒ I = I1 + I2


I1 =


Let 5x2 + 1 = t ….(i)


d(5x2 + 1) = dt


10x dx = dt …..(ii)


When x = 0; t = 5 × 02 + 1 = 1


When x = 1; t = 5 × 12 + 1 = 6


Substituting (i) and (ii) in I1


I1 = []


I1 =


I1 =


I2 = []


I2 =


I2 = 3/√5 tan-15


∵ I = I1 + I2


∴ I = 1/5 log 6 + 3/√5 tan-15


= 1/5 log 6 + 3/√5 tan-15



Question 15.

Evaluate


Answer:

Let I =

Put x2 = t ⇒ 2x dx = dt


When x = 0; t = 0


When x = 1; t = 1


Substituting t and dt in I


⇒ I = = � []


⇒ I =


= � (e – 1)



Question 16.

Evaluate


Answer:

Let I =

Dividing 5x2 by x2 + 4x + 3 we get 5 as quotient and –(20x + 15) as remainder


So, I =


⇒ I = =


⇒ I = 5 (2 – 1) -


⇒ I = 5 – I1


I1 =


Adding and subtracting 25 in the numerator


I1 =


I1 =


Let x2 + 4x + 3 = t


(2x + 4)dx = dt


∴ I1 =


I1 = 10 log t - []


I1 = 10 log t - []


I1 =


I1 = 10


I1 =


I1 =


I1 =


I1 =


I1 =


∵ I = 5 – I1


Substituting I1 in I we get


I = 5 –


= 5 –



Question 17.

Evaluate


Answer:

Let I =

⇒ I =


⇒ I = []


⇒ I = 2 (tan π/4 – tan 0) + � ((π/4)4 – 0) + 2 (π/4 – 0)


⇒ I =


⇒ I =




Question 18.

Evaluate


Answer:

Let I =

We know,


Substituting in I, we get


⇒ I = []


⇒ I =


= 0



Question 19.

Evaluate


Answer:

Let I =

I =


∴ I = I1 + I2


I1 =


Let x2 + 4 = t


2x dx = dt


When x = 0; t = 4


When x = 2; t = 22 + 4 = 8


Substituting t and dt in I1


⇒ I1 = []


⇒ I1 = 3 [log |8| - log |4|] = 3 log 8/4


⇒ I1 = 3 log � = -3 log 2


I2 = = []


⇒ I2 =


⇒ I2 = = 3π/8


Now I = I1 + I2


I = 3 log � + 3π/8


= 3 log � + 3π/8



Question 20.

Evaluate


Answer:

Let I =

⇒ I =


I = I1 + I2


I1 =


⇒ I1 = []


⇒ I1 =


⇒ I1 = e – e – 0 + 1


⇒ I1 = 1


I2 = []


⇒ I2 =


⇒ I2 = =


Since, I = I1 + I2


∴ I = 1 +


= 1 +



Question 21.

equals
A. π/3

B. 2π/3

C. π/6

D. π/12


Answer:

Let I =

⇒ I =


⇒ I = []


⇒ I = π/12


π/12


Question 22.

equals
A. π/6

B. π/12

C. π/24

D. π/4


Answer:

Let I =

⇒ I =


Taking 9 common from Denominator in I


⇒ I = []


⇒ I =


⇒ I =


⇒ I = = π/24


= π/24