Integrals
Class 12th Mathematics NCERT Exemplar Solution
- integrate 2x-1/2x+3 dx = x-log| (2x+3)^2 |+c Verify the following :…
- integrate 2x+3/x^2 + 3x dx = log|x^2 + 3x|+c Verify the following :…
- integrate (x^2 + 2) dx/x+1 Evaluate the following:
- integrate e^6logx-e^5logx/e^4logx-e^3logxdx Question has been changed Evaluate the…
- integrate (1+cosx)/x+sinxdx Evaluate the following:
- integrate dx/1+cosx Evaluate the following:
- integrate tan^2xsec^4xdx Evaluate the following:
- integrate sinx+cosx/root 1+sin2x dx Evaluate the following:
- integrate root 1+sinxdx Evaluate the following:
- integrate x/root x+1 dx (Hint: Put √x = z) Evaluate the following:…
- integrate root a+x/a-x Evaluate the following:
- integrate x^1/2/1+x^3/4 dx (Hint : Put x = z^4) Evaluate the following:…
- integrate root 1+x^2/x^4 dx Evaluate the following:
- integrate dx/root 16-9x^2 Evaluate the following:
- integrate dt/root 3t-2t^2 Evaluate the following:
- integrate 3x-1/root x^2 + 9 dx Evaluate the following:
- integrate root 5-2x+x^2 dx Evaluate the following:
- integrate x/x^4 - 1 dx Evaluate the following:
- integrate x^2/1-x^4 dx x^2 = t Evaluate the following:
- integrate root 2ax-x^2 dx Evaluate the following:
- integrate sin^-1x/(1-x^2)^3/2 dx Evaluate the following:
- integrate (cos5x+cos4x)/1-2cos3xdx Evaluate the following:
- integrate sin^6x+cos^6x/sin^2xcos^2xdx Evaluate the following:
- integrate root x/root a^3 - x^3 dx Evaluate the following:
- integrate cosx-cos2x/1-cosxdx Evaluate the following:
- integrate dx/x root x^4 - 1 (h : putx^2 = sectheta) Evaluate the following:…
- integrate _0^2 (x^2 + 3) dx Evaluate the following as limit of sums:…
- integrate _0^2e^x dx Evaluate the following as limit of sums:
- integrate _0^1 dx/e^x + e^-x Evaluate the following:
- integrate _0^pi /2 tanxdx/1+m^2tan^2x Evaluate the following:
- integrate _1^2 dx/root (x-1) (2-x) Evaluate the following:
- integrate _0^1 xdx/root 1+x^2 Evaluate the following:
- integrate _0^pi xsinxcos^2xdx Evaluate the following:
- integrate _0^1/2 dx/(1+x^2) root 1-x^2 (Hint: let x = sin θ) Evaluate the…
- integrate x^2 dx/x^4 - x^2 - 12 Evaluate the following:
- integrate x^2 dx/(x^2 + a^2) (x^2 + b^2) Evaluate the following:
- integrate _0^pi x/1+sinx Evaluate the following:
- integrate 2x-1/(x-1) (x+2) (x-3) dx Evaluate the following:
- integrate e^tan^-1x (1+x+x^2/1+x^2) dx Evaluate the following:
- integrate sin^-1root x/a+x dx (Hint: Put x = a tan^2 θ) Evaluate the following:…
- integrate _ pi /3^pi /2 root 1+cosx/(1-cos)^5/2 Evaluate the following:…
- integrate e^-3xcos^3xdx Evaluate the following:
- integrate root tanxdx (Hint: Put tan x = t^2) Evaluate the following:…
- integrate _0^pi /2 dx/(a^2cos^2x+b^2sin^2x)^2 (Hint: Divide Numerator and…
- integrate _0^1xlog (1+2x) dx Evaluate the following:
- integrate _0^pi xlogsinxdx Evaluate the following:
- integrate _ - pi /4^pi /4 log (sinx+cosx) dx Evaluate the following:…
- integrate cos2x-cos2theta /cosx-costheta dx is equal toA. 2(sinx + xcos θ) + C B.…
- dx/sin (x-a) sin (x-b) is equal toA. sin (b-a) log| sin (x-b)/sin (x-a) |+c B.…
- integrate tan^-1root xdx is equal toA. (x+1) tan^-1root x - root x+c B.…
- integrate e^x (1-x/1+x^2)^2 dx is equal toA. e^x/1+x^2 + c B. - e^x/1+x^2 + c C.…
- integrate x^9/(4x^2 + 1)^6 dx is equal toA. 1/5x (4 + 1/x^2)^-5 + c B. 1/5 (4 +…
- If integrate dx/(x+2) (x^2 + 1) = alog|1+x^2 |+btan^-1x + 1/5 log |x + 2| + C,…
- integrate x^3/x+1 is equal toA. x + x^2/2 + x^3/3-log|1-x|+c B. x + x^2/2 -…
- integrate x+sinx/1+cosxdx is equal toA. log |1 + cos x| + C B. log |x + sin x| +…
- If x^3 dx/root 1+x^2 = a (1+x^2)^3/2 + b root 1+x^2 + c , thenA. a = 1/3 , b = 1…
- integrate _ - pi /4^pi /4 dx/1+cos2x is equal toA. 1 B. 2 C. 3 D. 4…
- integrate _0^pi /2 root 1-sin2xdx is equal toA. 2√2 B. 2(√2 + 1) C. 2 D. 2 (√2…
- integrate _0^pi /2 cosxe^sinxdx is equal to ___________. Fill in the blanks in…
- integrate x+3/(x+4)^2 e^x dx = ___________. Fill in the blanks in each of the…
- If integrate _0^2 1/1+4x^2 dx = pi /8 then a = ____________. Fill in the blanks…
- integrate sinx/3+4cos^2xdx = __________. Fill in the blanks in each of the…
- The value of integrate _ - pi^pi sin^3xcos^2xdx is ____________. Fill in the…
Exercise
Question 1.Verify the following :
Answer:
To Verify; 
Let t = 2x + 3
Hence Verified
Question 2.
Verify the following :
Answer:
To Verify; 
Let; t=x2 + 3x
⇒ dt = 2x + 3
[∵t = x2 + 3x]
Hence Verified
Question 3.
Evaluate the following:
Answer:
Given; 
Let t = x + 1
⇒ dx = dt
Question 4.
Evaluate the following:
Question has been changed
Answer:
Given; 
As we know n log x = log xn
Take x3 common out of numerator and denominator to get,
= ∫ x2 dx
So,
Question 5.
Evaluate the following:
Answer:
Given; 
Let t = x + sin x
⇒ dt = (1 + cos x) dx
=log |t|+C
=log |x + sin x |+C
Question 6.
Evaluate the following:
Answer:
Given; 
As we know,
∫cosec x cot x dx=−cosec x+c
∫cosec2x dx=−cot x+c
Question 7.
Evaluate the following:
Answer:
Given; ∫tan2 x sec4 x dx
=∫ tan2 x sec2 x (1+ tan2 x) dx
Let; tan x = y
⇒ sec2 x dx = dy
=∫y2+y4 dy
Question 8.
Evaluate the following:
Answer:
Given; 
= x + C
Question 9.
Evaluate the following:
Answer:
Given; 
Question 10.
Evaluate the following:
(Hint: Put √x = z)
Answer:
Given; 
Let z = √x
⇒ x = z2
⇒ dx = 2z dz
Let; t = z + 1
i.e. t = √x + 1
⇒ dt = dz
Question 11.
Evaluate the following:
Answer:
Given; 
Let t=a2 -x2
⇒-2x dx=dt
and 
Question 12.
Evaluate the following:
(Hint : Put x = z4)
Answer:
Given; 
Let x = z4
⇒ dx = 4z3 dz
Let t = 1 + z3 [i.e. t = 1 + x3/4]
⇒ dt = 3z2 dz
Question 13.
Evaluate the following:
Answer:
Given; 
Let x = tan y
⇒ dx = sec2 y dy
Let t = sin y
⇒ dt = cos y dy
Question 14.
Evaluate the following:
Answer:
Given; 
Question 15.
Evaluate the following:
Answer:
Given; 
Question 16.
Evaluate the following:
Answer:
Given; 
[Let; x2 + 9 = y ⇒ 2x dx = dy]
Question 17.
Evaluate the following:
Answer:
Given; 
Question 18.
Evaluate the following:
Answer:
Given; 
[Let; t = x2⇒ dt = 2x dx]
Question 19.
Evaluate the following:
Answer:
Given; 
As we know,
Question 20.
Evaluate the following:
Answer:
Given; 
Question 21.
Evaluate the following:
Answer:
Given; 
Apply integration by parts
Question 22.
Evaluate the following:
Answer:
Given; 
Question 23.
Evaluate the following:
Answer:
Given; 
Question 24.
Evaluate the following:
Answer:
Given; 
Question 25.
Evaluate the following:
Answer:
Given; 
Question 26.
Evaluate the following:
Answer:
Given; 
Question 27.
Evaluate the following as limit of sums:
Answer:
Given; 
We know 
Here a = 0 , b = 2
⇒ nh = 2
Question 28.
Evaluate the following as limit of sums:
Answer:
We know
Here a = 0 , b = 2
⇒ nh = 2
= e2 - 1
Question 29.
Evaluate the following:
Answer:
Given; 
Question 30.
Evaluate the following:
Answer:
Given; 
By applying partial fraction;
When u = −1;
By applying the given limits 0 to π/2
Question 31.
Evaluate the following:
Answer:
Given 
⇒ 
Using perfect square method for the denominator
We know that 
=sin-1(1)-sin-1 (-1)
We know sin-1 (-θ ) = - sin θ
= π
Question 32.
Evaluate the following:
Answer:
Given 
Now put 
⇒ 2x dx=dt
At x=0, t=1 and
at x=1, t=2
Question 33.
Evaluate the following:
Answer:
Using Property
Let 
… (1)
⇒
⇒As sin(π-x) = sin x and
cos(π-x) = - cos x
⇒
⇒
Now let cos x=t
⇒- sin x dx=dt
And, at x=0, t=1
and at x=π, t=-1
⇒
⇒
⇒
⇒
Question 34.
Evaluate the following:
(Hint: let x = sin θ)
Answer:
Given 
⇒Let 
⇒
At x=0, θ=0
⇒
⇒
⇒
⇒
⇒As sec2 θ-tan2 θ=1
⇒
Now put tan θ=t
⇒sec2 θ dθ=dt
At θ=0, t=0
⇒
As 
⇒
⇒
Question 35.
Evaluate the following:
Answer:
Given: 
Put
⇒
(Concept of partial fraction)
On comparing coefficients of ‘t’ we get
⇒ 
⇒ Now put 
back in the above eq.
⇒ 
⇒ 
⇒ Now 
⇒ 
⇒ 
Question 36.
Evaluate the following:
Answer:
Given 
Put
⇒ 
⇒ 
(Concept of partial fraction)
⇒ 
On comparing coefficients of ‘t’ we get
⇒ 
⇒ 
⇒ Now put back in the above eq.
⇒ 
⇒ 
⇒ Now
⇒ 
⇒ 
⇒ 
Question 37.
Evaluate the following:
Answer:
Given 
Let 
Now using Property 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Put 
⇒ 
⇒ At 
⇒ 
⇒ 
⇒ 
Question 38.
Evaluate the following:
Answer:
Given: 
Using the concept of partial fractions,
⇒ 
⇒ 
Comparing coefficients:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 39.
Evaluate the following:
Answer:
Given: 
Put 
⇒ 
⇒ 
⇒ As 
⇒ 
⇒ 
Now using the property:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 40.
Evaluate the following:
(Hint: Put x = a tan2 θ)
Answer:
Given: 
⇒ 
⇒ 
⇒ 
⇒ As
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 41.
Evaluate the following:
Answer:
Given: 
Using Trigonometric identities:
⇒ cos2x=2 cos2x-1= 1-2 sin2 x
⇒ 
Question 42.
Evaluate the following:
Answer:
Given: 
Using trigonometric identity 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 43.
Evaluate the following:
(Hint: Put tan x = t2)
Answer:
Given 
Put 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Taking out 
common in both the numerators
⇒ 
⇒ 
⇒ Now 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Put (2) and (3) in (1)
⇒ 
⇒ 
⇒ 
⇒ 
Question 44.
Evaluate the following:
(Hint: Divide Numerator and Denominator by cos4x)
Answer:
Given: 
Dividing Numerator and Denominator by cos4x
⇒ 
⇒ 
⇒ 
⇒ Put 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 45.
Evaluate the following:
Answer:
Given 
Let 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ Apply Integration by parts
⇒ 
⇒ 
⇒ Put (2) and (3) in (1)
⇒ 
⇒ 
Question 46.
Evaluate the following:
Answer:
Given:
Using the property:
Let 
⇒ 
-
⇒ 
⇒ 
⇒ 
Using the property: 
⇒ 
⇒ Let 
..(3)
⇒ Using the property:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 2dx=dt and limits changes from 0to π
⇒ 
⇒ from equation (2)
again becomes,
⇒ 
⇒ From eq. (3)
⇒ 
⇒ 
..(7)
⇒ On putting (7) in (2) and the obtained result in(1)
⇒ 
⇒ 
Question 47.
Evaluate the following:
Answer:
Given: 
Let 
Using the property:
⇒ 
⇒ 
⇒ 
Adding equation (1) and (2)
⇒ 
⇒ 
⇒ Put 2x=t
⇒ 2xdx=dt
⇒ 
⇒ As cos(-x) = cos x
Using property: 
⇒ 
Using the property:
⇒ 
⇒ Now From previous question eq.(7) we obtained
⇒ 
⇒ 
Question 48.
is equal to
A. 2(sinx + xcos θ) + C
B. 2(sinx – xcos θ) + C
C. 2(sinx + 2xcos θ) + C
D. 2(sinx – 2x cos θ) + C
Answer:
Using Trigonometric identity 
⇒ 
=2∫(cos x +cos θ)dx
=2∫cosxdx+2∫cosθdx
⇒ 
Question 49.
is equal to
A. 
B. 
C. 
D. 
Answer:
Given:
Multiply Nr and Dr by 
⇒ 
⇒ 
⇒ Also
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 50.
is equal to
A. 
B. 
C. 
D. 
Answer:
Given: 
Put 
⇒ 
⇒ 
⇒ 
⇒ Now apply integration by part on ∫t tan-1 t dt
⇒
⇒ 
⇒
⇒
⇒ Put (3) in (2) and the resulting equation in (1)
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 51.
is equal to
A. 
B. 
C. 
D. 
Answer:
Given:
⇒ 
⇒ 
Now using the property:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 52.
is equal to
A. 
B. 
C. 
D. 
Answer:
Given: 
Taking x2 out from the denominator
⇒ 
⇒ 
⇒ Now put 
⇒ 
⇒ 
⇒ 
Question 53.
If 
log |x + 2| + C, then
A. 
B. 
C. 
D. 
Answer:
Given: 
…(1)
Using concept of partial fractions
⇒ 
⇒ 
⇒ 
⇒ A+B=0 …(1)
⇒ C+2B=0 …(2)
⇒ A+2C=1 …(3)
⇒ On solving the above three equations we get
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
On comparing (1) and (2) we get,
⇒ 
Question 54.
is equal to
A. 
B. 
C. 
D. 
Answer:
Given: 
⇒ 
⇒ 
⇒ 
⇒ 
Question 55.
is equal to
A. log |1 + cos x| + C
B. log |x + sin x| + C
C. 
D. 
Answer:
D.
Given: 
As we know
∴ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ ∫(2t+2t tan2 t+2 tan t) dt=2∫(t + t tan^2 t+ tan t) dt
⇒ 2∫tdt+2∫t (sec2(t-1)dt+2∫tan t dt
⇒ 2∫tdt+2∫t sec2 t dt-2∫tdt+2∫tan t dt
⇒ 2∫t sec2 t dt+2∫tant dt ….(1)
Applying Integration by parts on ∫t sec2 t dt
⇒ 
⇒ ∫t sec2 t dt=t tan t-∫tan t dt ….(2)
⇒ Put (2) in(1)
⇒ 
⇒ 
Question 56.
If 
, then
A. 
B. 
C. 
D. 
Answer:
Given: 
…(1)
Put
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
…(3)
Comparing (1) and (3)
⇒ 
Question 57.
is equal to
A. 1
B. 2
C. 3
D. 4
Answer:
Given:
Using trigonometric identities:
Question 58.
is equal to
A. 2√2
B. 2(√2 + 1)
C. 2
D. 2 (√2 –1)
Answer:
As 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ On solving the Above Integral we get 
Question 59.
Fill in the blanks in each of the following
is equal to ___________.
Answer:
e-1
Given: 
Put 
⇒ 
⇒ At 
⇒ 
Question 60.
Fill in the blanks in each of the following
___________.
Answer:
Given 
⇒ 
⇒ 
⇒ 
Now using the property:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 61.
Fill in the blanks in each of the following
If 
then a = ____________.
Answer:
Given: 
Question 62.
Fill in the blanks in each of the following
__________.
Answer:
Given 
⇒ 
⇒
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 63.
Fill in the blanks in each of the following
The value of 
is ____________.
Answer:
0
Using the property:
Let 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 