NCERT Solutions For Class 12 Maths Chapter 7 Integrals Ex 7.10
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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 7.10 |
Number of Questions Solved | 10 |

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.10
NCERT TEXTBOOK EXERCISES
Ex 7.10 Class 12 Maths Question 1.
$\int _{ 0 }^{ 1 }{ \frac { x }{ { x }^{ 2 }+1 } } dx=I$
Solution:
Let x² + 1 = t
⇒2xdx = dt
when x = 0, t = 1 and when x = 1, t = 2
Ex 7.10 Class 12 Maths Question 2.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sqrt { sin\phi } { cos }^{ 5 }\phi d\phi =I }$
Solution:
$I=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sqrt { sin\phi } { (1-{ sin }^{ 2 }) }^{ 2 }cos\phi d\phi }$
put sinφ = t,so that cosφdφ = dt

Ex 7.10 Class 12 Maths Question 3.
$\int _{ 0 }^{ 1 }{ { sin }^{ -1 } } \left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) dx=I$
Solution:
let x = tanθ =>dx = sec²θ dθ
when x = 0 => θ = 0
and when x = 1 => θπ/4
1/2

Ex 7.10 Class 12 Maths Question 4.
$\int _{ 0 }^{ 2 }{ x\sqrt { x+2 } } dx=I(say)(put\quad x+2={ t }^{ 2 })$
Solution:
let x+2 = t =>dx = dt
when x = 0,t = 2 and when x = 2, t = 4

Ex 7.10 Class 12 Maths Question 5.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx\quad dx }{ 1+{ cos }^{ 2 }x } =I }$
Solution:
put cosx = t
so that -sinx dx = dt
when x = 0, t = 1; when x=π2, t = 0
Undefined control sequence \therefore
Ex 7.10 Class 12 Maths Question 6.
$\int _{ 0 }^{ 2 }{ \frac { dx }{ x+4-{ x }^{ 2 } } =I }$
Solution:
$\int _{ 0 }^{ 2 }{ \frac { dx }{ x+4-{ x }^{ 2 } } =I }$

Ex 7.10 Class 12 Maths Question 7.
$\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } =I }$
Solution:
$I=\int { -1 }^{ 1 }{ \frac { dx }{ { (x+1) }^{ 2 }+{ 2 }^{ 2 } } } =\frac { 1 }{ 2 } { \left[ { tan }^{ -1 }\frac { x+1 }{ 2 } \right] }{ -1 }^{ 1 }\quad =\frac { \pi }{ 8 }$
Ex 7.10 Class 12 Maths Question 8.
$\int _{ 1 }^{ 2 }{ \left[ \frac { 1 }{ x } -\frac { 1 }{ { 2x }^{ 2 } } \right] { e }^{ 2x }dx } =I$
Solution:
let 2x = t ⇒ 2dx = dt
when x = 1, t = 2 and when x = 2, t = 4
$I=\int { 2 }^{ 4 }{ e } ^{ t }\left( \frac { 1 }{ t } -\frac { 1 }{ { t }^{ 2 } } \right) dt\quad ={ e }^{ t }{ \left[ \frac { 1 }{ t } \right] }{ 2 }^{ 4 }\quad =\frac { e^{ 2 } }{ 2 } \left[ \frac { { e }^{ 2 } }{ 2 } -1 \right]$
Choose the correct answer in Exercises 9 and 10
Ex 7.10 Class 12 Maths Question 9.
The value of integral $\int _{ \frac { 1 }{ 3 } }^{ 1 }{ \frac { { { (x-x }^{ 3 }) }^{ \frac { 1 }{ 3 } } }{ { x }^{ 4 } } dx }$ is
(a) 6
(b) 0
(c) 3
(d) 4
Solution:
(a) let I = $\int { \frac { 1 }{ 3 } }^{ 1 }{ \frac { { { (x-x }^{ 3 }) }^{ \frac { 1 }{ 3 } } }{ { x }^{ 4 } } dx } \quad =\int { \frac { 1 }{ 3 } }^{ 1 }{ \frac { { x }^{ \frac { 1 }{ 3 } }(1-{ x }^{ 2 })^{ \frac { 1 }{ 3 } } }{ { x }^{ 4 } } dx }$

Ex 7.10 Class 12 Maths Question 10.
$If\quad f(x)=\int _{ 0 }^{ x }{ tsint,\quad then\quad { f }^{ \prime }(x)\quad is }$
(a) cosx+xsinx
(b) xsinx
(c) xcosx
(d) sinx+xcosx
Solution:
(b) $f(x)=\int _{ 0 }^{ x }{ tsint\quad dt }$
$=t(-cost)-\int { 1{ \left[ (-cost)dt \right] }_{ 0 }^{ x } }$
=-x cox+sinx
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.10 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