# NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 Questions and Answers are prepared by our highly skilled subject experts.

## NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.11

Question 1.
$$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 }x\quad dx }$$
Solution:

Question 2.
$$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx }$$
Solution:

Question 3.
$$\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x d x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}$$
Solution:

Question 4.
$$\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{5} x d x}{\sin ^{5} x+\cos ^{5} x}$$
Solution:

Question 5.
$$\int_{-5}^{5}|x+2| d x$$
Solution:
|x+2| = x + 2 if x + 2 ≥ 0 ⇒ x ≥ – 2
|x+2| = -(x + 2) if x + 2 < 0 ⇒ x < – 2

Question 6.
$$\int_{2}^{8}|x-5| d x$$
Solution:

Question 7.
$$\int _{ 0 }^{ 1 }{ x(1-x)^{ n }dx } =I$$
Solution:

Question 8.
$$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ log(1+tanx)dx }$$
Solution:

Question 9.
$$\int_{0}^{2} x \sqrt{2-x} d x$$
Solution:

Question 10.
$$\int_{0}^{\frac{\pi}{2}}(2 \log \sin x-\log \sin 2 x)$$dx
Solution:

Question 11.
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x d x$$
Solution:
Here f(x) = sin²x
∴ f(x) = sin²(-x) = [sin(-x)]²
= (-sinx)² = sin²x
Since f(-x) = f(x),
f(x) = sin²x is an even function.

Question 12.
$$\int_{0}^{\pi} \frac{x d x}{1+\sin x}$$
Solution:

Question 13.
$$\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 7 } } xdx$$
Solution:
Let f(x) = sin7 xdx
⇒ f(-x) = -sin7 x = -f(x)
⇒ f(x) is an odd function of x
⇒ $$\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 7 } } xdx=0$$

Question 14.
$$\int _{ 0 }^{ 2\pi }{ { cos }^{ 5 } } xdx$$
Solution:
Let f(x) = cos5 x
f(2π – x) = cos5(2π – x) = cos5 x
f(2π – x) = f(x)

Question 15.
$$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx-cosx }{ 1+sinx\quad cosx } dx }$$
Solution:

Question 16.
$$\int_{0}^{\pi} \log (1+\cos x) d x$$
Solution:

Question 17.
$$\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} d x$$
Solution:

Question 18.
$$\int_{0}^{4}|x-1| d x$$
Solution:

Question 19.
Show that $$\int_{0}^{a} f(x) g(x) d x=2 \int_{0}^{a} f(x) d x$$ if f and g are defined as f(x) = f(a – x) and g(x) + g(a – x) = 4
Solution:

Question 20.
The value of
$$\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \left( { x }^{ 3 }+xcosx+{ tan }^{ 5 }x+1 \right) dx }$$ is
(a) 0
(b) 2
(c) π
(d) 1
Solution:
(c) π
Let f(x) = x³ + x cosx + tan5x
f(- x) = (- x)³ + (- x)cos(- x) + tan5(- x)
= – x³ – x cosx – tan5x
= – f(x)
∴ f(x) is an odd function

Question 21.
The value of $$\int_{0}^{\frac{x}{2}} \log \left(\frac{4+3 \sin x}{4+3 \cos x}\right) d x$$ is
(a) 2
(b) $$\frac { 3 }{ 4 }$$
(c) 0
(d) – 2
Solution:

I = – 1
2I = 0 Hence I = 0
Evaluate the following.

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