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

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

Question 1.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 }x\quad dx }\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 1

Question 2.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx } \)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 2

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:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 3

Question 4.
\(\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{5} x d x}{\sin ^{5} x+\cos ^{5} x}\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 4

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
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 5

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

Question 6.
\(\int_{2}^{8}|x-5| d x\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 5a

Question 7.
\(\int _{ 0 }^{ 1 }{ x(1-x)^{ n }dx } =I\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 6

Question 8.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ log(1+tanx)dx } \)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 7

Question 9.
\(\int_{0}^{2} x \sqrt{2-x} d x\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 8

Question 10.
\(\int_{0}^{\frac{\pi}{2}}(2 \log \sin x-\log \sin 2 x)\)dx
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 9

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

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.
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 10

Question 12.
\(\int_{0}^{\pi} \frac{x d x}{1+\sin x}\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 11

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

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

Question 15.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx-cosx }{ 1+sinx\quad cosx } dx } \)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 13

Question 16.
\(\int_{0}^{\pi} \log (1+\cos x) d x\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 14

Question 17.
\(\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} d x\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 15

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

Question 18.
\(\int_{0}^{4}|x-1| d x\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 16

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:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 17

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
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 18

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

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:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 19
I = – 1
2I = 0 Hence I = 0
Evaluate the following.

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