NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11.

Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 7
Chapter Name	Integrals
Exercise	Ex 7.11
Number of Questions Solved	21
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.

Question 1.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 }x\quad dx } =I$
Solution:
$I=\frac { 1 }{ 2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ (1+cos2x)dx } =\frac { 1 }{ 2 } { \left[ x+\frac { sin2x }{ 2 } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }\quad =\frac { \pi }{ 4 }$

Question 2.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx }$
Solution:
let I = $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 2

Question 3.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { sin }^{ \frac { 3 }{ 2 } }xdx }{ { sin }^{ \frac { 3 }{ 2 } }x+{ cos }^{ \frac { 3 }{ 2 } }dx } dx }$
Solution:
let I = $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { sin }^{ \frac { 3 }{ 2 } }xdx }{ { sin }^{ \frac { 3 }{ 2 } }x+{ cos }^{ \frac { 3 }{ 2 } }dx } dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 3

Question 4.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 5 }xdx }{ { sin }^{ 5 }x+{ cos }^{ 5 }x } }$
Solution:
let I = $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 5 }xdx }{ { sin }^{ 5 }x+{ cos }^{ 5 }x } }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 4

Question 5.
$\int _{ -5 }^{ 5 }{ \left| x+2 \right| dx=I }$
Solution:
$I=\int _{ -5 }^{ 5 }{ \left| x+2 \right| dx+\int _{ -2 }^{ 5 }{ \left| x+2 \right| dx } }$
at x = – 5, x + 2 < 0; at x = – 2, x + 2 = 0; at x = 5, x + 2>0;x + 2<0, x + 2 = 0, x + 2>0
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 5

Question 6.
$\int _{ 2 }^{ 8 }{ |x-5|dx } =I$
Solution:
$\int _{ 2 }^{ 8 }{ |x-5|dx } =I$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 6

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

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

Question 9.
$\int _{ 0 }^{ 2 }{ x\sqrt { 2-x } dx=I }$
Solution:
let 2-x = t
⇒ – dx = dt
when x = 0, t = 2 and when x = 2,t = 0
$\frac { 1 }{ 2 }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 9.

Question 10.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2logsinx-logsin2x \right) dx=I }$
Solution:
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2logsinx-logsin2x \right) dx=I }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 10

Question 11.
$\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 } } xdx$
Solution:
Let f(x) = sin² x
f(-x) = sin² x = f(x)
∴ f(x) is an even function
$\therefore \int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 } } xdx\quad =2\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left[ \frac { 1-cos2x }{ 2 } \right] dx }$
$={ \left[ x-\frac { sin2x }{ x } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }\therefore I=\frac { \pi }{ 2 }$

Question 12.
$\int _{ 0 }^{ \pi }{ \frac { xdx }{ 1+sinx } }$
Solution:
let I = $\int _{ 0 }^{ \pi }{ \frac { xdx }{ 1+sinx } }$ …(i)
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 12

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 x

NCERT Solutions for Class 12 Maths Chapter 7 Integrals 14.1

Question 15.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx-cosx }{ 1+sinx\quad cosx } dx }$
Solution:
let I = $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx-cosx }{ 1+sinx\quad cosx } dx }$ …(i)
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 15

Question 16.
$\int _{ 0 }^{ \pi }{ log(1+cosx)dx }$
Solution:
let I = $\int _{ 0 }^{ \pi }{ log(1+cosx)dx }$
then I = $\int _{ 0 }^{ \pi }{ log[1+cos(\pi -x)]dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 16

Question 17.
$\int _{ 0 }^{ a }{ \frac { \sqrt { x } }{ \sqrt { x } +\sqrt { a-x } } dx }$
Solution:
let I = $\int _{ 0 }^{ a }{ \frac { \sqrt { x } }{ \sqrt { x } +\sqrt { a-x } } dx }$ …(i)
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 17

Question 18.
$\int _{ 0 }^{ 4 }{ \left| x-1 \right| dx=I }$
Solution:
$I=-\int _{ 0 }^{ 1 }{ (x-1)dx } +\int _{ 1 }^{ 4 }{ (x-1)dx }$
$=-{ \left[ \frac { { x }^{ 2 } }{ 2 } -x \right] }_{ 0 }^{ 1 }+{ \left[ \frac { { x }^{ 2 } }{ 2 } -x \right] }_{ 1 }^{ 4 }=5$

Question 19.
show that $4\int _{ 0 }^{ a }{ f(x)g(x)dx } =2\int _{ 0 }^{ a }{ f(x)dx }$ if f and g are defined as f(x)=f(a-x) and g(x)+g(a-x)=4
Solution:
let I = $\int _{ 0 }^{ a }{ f(x)g(x)dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 19

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) let I = $\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \left( { x }^{ 3 }+xcosx+{ tan }^{ 5 }x+1 \right) dx }$ is
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 20

Question 21.
The value of $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\left[ \frac { 4+3sinx }{ 4+3sinx } \right] dx }$ is
(a) 2
(b) $\frac { 3 }{ 4 }$
(c) 0
(d) -2
Solution:
let I = $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\left[ \frac { 4+3sinx }{ 4+3sinx } \right] dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 21

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NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

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