NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9.
- Integrals Class 12 Ex 7.1
- Integrals Class 12 Ex 7.2
- Integrals Class 12 Ex 7.3
- Integrals Class 12 Ex 7.4
- Integrals Class 12 Ex 7.5
- Integrals Class 12 Ex 7.6
- Integrals Class 12 Ex 7.7
- Integrals Class 12 Ex 7.8
- Integrals Class 12 Ex 7.10
- Integrals Class 12 Ex 7.11
| Board | CBSE |
| Textbook | NCERT |
| Class | Class 12 |
| Subject | Maths |
| Chapter | Chapter 7 |
| Chapter Name | Integrals |
| Exercise | Ex 7.9 |
| Number of Questions Solved | 22 |
| Category | NCERT Solutions |
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9
Evaluate the definite integrals in Exercise 1 to 20.
Ex 7.9 Class 12 Maths Question 1.
\(\int _{ -1 }^{ 1 }{ { (x+1 }) } dx\quad \)
Solution:
\({ =\left[ \frac { { x }^{ 2 } }{ 2 } +x \right] }_{ -1 }^{ 1 }=\frac { 1 }{ 2 } (1-1)+(1+1)\quad =2\)
Ex 7.9 Class 12 Maths Question 2.
\(\int _{ 2 }^{ 3 }{ \frac { 1 }{ x } dx } \)
Solution:
\(={ \left[ log\quad x \right] }_{ 2 }^{ 3 }\quad =log3-log2\quad =log\frac { 3 }{ 2 } \)
Ex 7.9 Class 12 Maths Question 3.
\(\int _{ 1 }^{ 2 }{ \left( { 4x }^{ 3 }-{ 5x }^{ 2 }+6x+9 \right) dx } \)
Solution:
\(={ \left[ \frac { { 4x }^{ 4 } }{ 4 } -\frac { { 5x }^{ 3 } }{ 3 } +\frac { { 6x }^{ 2 } }{ 2 } +9x \right] }_{ 1 }^{ 2 }\)
\(={ \left[ { x }^{ 4 }-\frac { 5 }{ 3 } { x }^{ 3 }+{ 3x }^{ 2 }+9x \right] }_{ 1 }^{ 2 }\quad =\frac { 64 }{ 3 } \)
Ex 7.9 Class 12 Maths Question 4.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ sin2x\quad dx } \)
Solution:
\(={ \left[ -\frac { 1 }{ 2 } cos2x \right] }_{ 0 }^{ \frac { \pi }{ 4 } }\quad =\frac { 1 }{ 2 } \)
Ex 7.9 Class 12 Maths Question 5.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ cos2x\quad dx } \)
Solution:
\(={ \left[ \frac { 1 }{ 2 } sin2x \right] }_{ 0 }^{ \frac { \pi }{ 2 } }\quad =0\)
Ex 7.9 Class 12 Maths Question 6.
\(\int _{ 4 }^{ 5 }{ { e }^{ x }dx } \)
Solution:
\(={ \left[ { e }^{ x } \right] }_{ 4 }^{ 5 }\quad ={ e }^{ 5 }-{ e }^{ 4 }\)
Ex 7.9 Class 12 Maths Question 7.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ tanx\quad dx } \)
Solution:
\(={ \left[ log\quad secx \right] }_{ 0 }^{ \frac { \pi }{ 4 } }\quad =\frac { 1 }{ 2 } log2\)
Ex 7.9 Class 12 Maths Question 8.
\(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 4 } }{ cosec\quad xdx } \)
Solution:
\(=log{ \left( cosecx-cotx \right) }_{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 4 } }\)
\(=log(\sqrt { 2 } -1)-log(2-\sqrt { 3 } )\quad =log\left( \frac { \sqrt { 2 } -1 }{ 2-\sqrt { 3 } } \right) \)
Ex 7.9 Class 12 Maths Question 9.
\(\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 1-{ x }^{ 2 } } } } \)
Solution:
\(={ sin }^{ -1 }(1)-{ sin }^{ -1 }(0)\quad =\frac { \pi }{ 2 } \)
Ex 7.9 Class 12 Maths Question 10.
\(\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } } \)
Solution:
\(={ \left[ { tan }^{ -1 }x \right] }_{ 0 }^{ 1 }\quad ={ tan }^{ -1 }(1)-{ ta }n^{ -1 }(0)\quad =\frac { \pi }{ 4 } \)
Ex 7.9 Class 12 Maths Question 11.
\(\int _{ 2 }^{ 3 }{ \frac { dx }{ { x }^{ 2 }-1 } } \)
Solution:
\(={ \left[ \frac { 1 }{ 2 } log\left( \frac { x-1 }{ x+1 } \right) \right] }_{ 2 }^{ 3 }\quad =\frac { 1 }{ 2 } log\frac { 3 }{ 2 } \)
Ex 7.9 Class 12 Maths Question 12.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 } } xdx\)
Solution:
\(=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { \frac { 1+cos2x }{ 2 } } } dx=\frac { 1 }{ 2 } { \left[ x+\frac { sin2x }{ 2 } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }=\frac { \pi }{ 4 } \)
Ex 7.9 Class 12 Maths Question 13.
\(\int _{ 2 }^{ 3 }{ \frac { x }{ { x }^{ 2 }+1 } } dx\)
Solution:
\(=\frac { 1 }{ 2 } \int _{ 2 }^{ 3 }{ \frac { 2x }{ { x }^{ 2 }+1 } } dx\quad =\frac { 1 }{ 2 } { \left[ log\left( { x }^{ 2 }+1 \right) \right] }_{ 2 }^{ 3 }\quad =\frac { 1 }{ 2 } log2\)
Ex 7.9 Class 12 Maths Question 14.
\(\int _{ 0 }^{ 1 }{ \frac { 2x+3 }{ { 5x }^{ 2 }+1 } dx } \)
Solution:
\(=\frac { 1 }{ 5 } \int _{ 0 }^{ 1 }{ \frac { 10x }{ { 5x }^{ 2 }+1 } dx } +\frac { 3 }{ 5 } \int _{ 0 }^{ 1 }{ \frac { dx }{ { { x }^{ 2 }+\left[ \frac { 1 }{ \sqrt { 5 } } \right] }^{ 2 } } } \)
Ex 7.9 Class 12 Maths Question 15.
\(\int _{ 0 }^{ 1 }{ { xe }^{ { x }^{ 2 } }dx } \)
Solution:
let x² = t ⇒ 2xdx = dt
when x = 0, t = 0 & when x = 1,t = 1
\(\therefore I=\frac { 1 }{ 2 } \int _{ 0 }^{ 1 }{ { e }^{ t }dt } \quad =\frac { 1 }{ 2 } { \left( { e }^{ t } \right) }_{ 0 }^{ 1 }\quad =\frac { 1 }{ 2 } [e-1]\)
Ex 7.9 Class 12 Maths Question 16.
\(\int _{ 1 }^{ 2 }{ \frac { { 5x }^{ 2 } }{ { x }^{ 2 }+4x+3 } dx } \)
Solution:
\(\int _{ 1 }^{ 2 }{ \left( 5-\frac { 20x+15 }{ { x }^{ 2 }+4x+3 } \right) dx } \)
Ex 7.9 Class 12 Maths Question 17.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( { 2sec }^{ 2 }x+{ x }^{ 3 }+2 \right) dx } \)
Solution:
\(={ \left[ 2tanx+\frac { { x }^{ 4 } }{ 4 } +2x \right] }_{ 0 }^{ \frac { \pi }{ 4 } }\)
Ex 7.9 Class 12 Maths Question 18.
\(\int _{ 0 }^{ \pi }{ \left( { sin }^{ 2 }\frac { x }{ 2 } -{ cos }^{ 2 }\frac { x }{ 2 } \right) } dx\)
Solution:
\(=-\int _{ 0 }^{ \pi }{ cosx } dx\quad =-{ \left[ sinx \right] }_{ 0 }^{ \pi }-(0-0)\quad =0\)
Ex 7.9 Class 12 Maths Question 19.
\(\int _{ 0 }^{ 2 }{ \frac { 6x+3 }{ { x }^{ 2 }+4 } } dx\)
Solution:
\(=\int _{ 0 }^{ 2 }{ \frac { 6x }{ { x }^{ 2 }+4 } } dx+\int _{ 0 }^{ 2 }{ \frac { 3 }{ { x }^{ 2 }+4 } dx } \)
Ex 7.9 Class 12 Maths Question 20.
\(\int _{ 0 }^{ 1 }{ \left( { xe }^{ x }+sin\frac { \pi x }{ 4 } \right) dx } \)
Solution:
\(=\int _{ 0 }^{ 1 }{ { xe }^{ x }dx } +\int _{ 0 }^{ 1 }{ sin\frac { \pi x }{ 4 } } dx\)
Ex 7.9 Class 12 Maths Question 21.
\(\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ { 1+x }^{ 2 } } \quad equals } \)
(a) \(\frac { \pi }{ 3 } \)
(b) \(\frac { 2\pi }{ 3 } \)
(c) \(\frac { \pi }{ 6 } \)
(d) \(\frac { \pi }{ 12 } \)
Solution:
(d) \(\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ { 1+x }^{ 2 } } } \quad ={ \left[ { tan }^{ -1 }x \right] }_{ 1 }^{ \sqrt { 3 } }\quad =\frac { \pi }{ 3 } -\frac { \pi }{ 4 } \quad =\frac { \pi }{ 12 } \)
Ex 7.9 Class 12 Maths Question 22.
\(\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ 4+{ 9x }^{ 2 } } equals } \)
(a) \(\frac { \pi }{ 6 }\)
(b) \(\frac { \pi }{ 12 }\)
(c) \(\frac { \pi }{ 24 }\)
(d) \(\frac { \pi }{ 4 }\)
Solution:
(c) \(\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ 4+{ 9x }^{ 2 } } } \quad =\frac { 1 }{ 9 } \int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ { \left( \frac { 2 }{ 3 } \right) }^{ 2 }+{ x }^{ 2 } } } \)
\(=\frac { 1 }{ 6 } { \left[ { tan }^{ -1 }\left( \frac { 3x }{ 2 } \right) \right] }_{ 0 }^{ \frac { 2 }{ 3 } }\quad =\frac { 1 }{ 6 } \times \frac { \pi }{ 4 } \quad =\frac { \pi }{ 24 } \)
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