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

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11
NCERT TEXTBOOK EXERCISES
Ex 7.11 Class 12 Maths Question 1.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 }x\quad dx } =I$
Solution:
$I=\frac { 1 }{ 2 } \int { 0 }^{ \frac { \pi }{ 2 } }{ (1+cos2x)dx } =\frac { 1 }{ 2 } { \left[ x+\frac { sin2x }{ 2 } \right] }{ 0 }^{ \frac { \pi }{ 2 } }\quad =\frac { \pi }{ 4 }$
Ex 7.11 Class 12 Maths Question 2.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx }$
Solution:
let I = $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx }$

Ex 7.11 Class 12 Maths 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 = $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { sin }^{ \frac { 3 }{ 2 } }xdx }{ { sin }^{ \frac { 3 }{ 2 } }x+{ cos }^{ \frac { 3 }{ 2 } }dx } dx }$

Ex 7.11 Class 12 Maths Question 4.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 5 }xdx }{ { sin }^{ 5 }x+{ cos }^{ 5 }x } }$
Solution:
let I = $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 5 }xdx }{ { sin }^{ 5 }x+{ cos }^{ 5 }x } }$

Ex 7.11 Class 12 Maths 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

Ex 7.11 Class 12 Maths Question 6.
$\int _{ 2 }^{ 8 }{ |x-5|dx } =I$
Solution:
$\int _{ 2 }^{ 8 }{ |x-5|dx } =I$

Ex 7.11 Class 12 Maths Question 7.
$\int _{ 0 }^{ 1 }{ x(1-x)^{ n }dx } =I$
Solution:
$\int _{ 0 }^{ 1 }{ x(1-x)^{ n }dx } =I$

Ex 7.11 Class 12 Maths Question 8. $\int _{ 0 }^{ \frac { \pi }{ 4 } }{ log(1+tanx)dx }$
Solution:
let I = $\int _{ 0 }^{ \frac { \pi }{ 4 } }{ log(1+tanx)dx }$

Ex 7.11 Class 12 Maths 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
12

Ex 7.11 Class 12 Maths Question 10.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2logsinx-logsin2x \right) dx=I }$
Solution:
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2logsinx-logsin2x \right) dx=I }$

Ex 7.11 Class 12 Maths 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
Ex 7.11 Class 12 Maths Question 12.
$\int _{ 0 }^{ \pi }{ \frac { xdx }{ 1+sinx } }$
Solution:
let I = $\int _{ 0 }^{ \pi }{ \frac { xdx }{ 1+sinx } }$…(i)

Ex 7.11 Class 12 Maths 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
Ex 7.11 Class 12 Maths Question 14.
$\int _{ 0 }^{ 2\pi }{ { cos }^{ 5 } } xdx$
Solution:
let f(x) = cos5 x
⇒ f(2π – x) = cos5 x

Ex 7.11 Class 12 Maths 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)

Ex 7.11 Class 12 Maths Question 16.
$\int _{ 0 }^{ \pi }{ log(1+cosx)dx }$
Solution:
let I = $\int _{ 0 }^{ \pi }{ log(1+cosx)dx }$
then I = $\int _{ 0 }^{ \pi }{ log[1+cos(\pi -x)]dx }$


Ex 7.11 Class 12 Maths 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)

Ex 7.11 Class 12 Maths Question 18.
$\int _{ 0 }^{ 4 }{ \left| x-1 \right| dx=I }$
Solution:
$I=-\int { 0 }^{ 1 }{ (x-1)dx } +\int { 1 }^{ 4 }{ (x-1)dx }$
$=-{ \left[ \frac { { x }^{ 2 } }{ 2 } -x \right] }{ 0 }^{ 1 }+{ \left[ \frac { { x }^{ 2 } }{ 2 } -x \right] }{ 1 }^{ 4 }=5$
Ex 7.11 Class 12 Maths 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 = $\int _{ 0 }^{ a }{ f(x)g(x)dx }$

Ex 7.11 Class 12 Maths 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

Ex 7.11 Class 12 Maths Question 21.
The value of $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\left[ \frac { 4+3sinx }{ 4+3sinx } \right] dx }$ is
(a) 2
(b) 34
(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 Exercise 7.11 PDF
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Other Chapter of Class 12 Maths Chapter 7 Integrals
- Class 12 Maths Chapter 7 Integrals Ex 7.1
- Class 12 Maths Chapter 7 Integrals Ex 7.2
- Class 12 Maths Chapter 7 Integrals Ex 7.3
- Class 12 Maths Chapter 7 Integrals Ex 7.4
- Class 12 Maths Chapter 7 Integrals Ex 7.5
- Class 12 Maths Chapter 7 Integrals Ex 7.6
- Class 12 Maths Chapter 7 Integrals Ex 7.7
- Class 12 Maths Chapter 7 Integrals Ex 7.8
- Class 12 Maths Chapter 7 Integrals Ex 7.9
- Class 12 Maths Chapter 7 Integrals Ex 7.10
- Class 12 Maths Chapter 7 Integrals Ex 7.11