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:
$=\frac { 1 }{ 5 } \int _{ 0 }^{ 1 }{ \frac { 10x }{ { 5x }^{ 2 }+1 } dx } +\frac { 3 }{ 5 } \int _{ 0 }^{ 1 }{ \frac { dx }{ { { x }^{ 2 }+\left[ \frac { 1 }{ \sqrt { 5 } } \right] }^{ 2 } } }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9 Q14.1

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:
$\int _{ 1 }^{ 2 }{ \left( 5-\frac { 20x+15 }{ { x }^{ 2 }+4x+3 } \right) dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9 Q16.1

Ex 7.9 Class 12 Maths Question 17.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( { 2sec }^{ 2 }x+{ x }^{ 3 }+2 \right) dx }$
Solution:
$={ \left[ 2tanx+\frac { { x }^{ 4 } }{ 4 } +2x \right] }_{ 0 }^{ \frac { \pi }{ 4 } }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9 Q17.1

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:
$=\int _{ 0 }^{ 2 }{ \frac { 6x }{ { x }^{ 2 }+4 } } dx+\int _{ 0 }^{ 2 }{ \frac { 3 }{ { x }^{ 2 }+4 } dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9 Q19.1

Ex 7.9 Class 12 Maths Question 20.
$\int _{ 0 }^{ 1 }{ \left( { xe }^{ x }+sin\frac { \pi x }{ 4 } \right) dx }$
Solution:
$=\int _{ 0 }^{ 1 }{ { xe }^{ x }dx } +\int _{ 0 }^{ 1 }{ sin\frac { \pi x }{ 4 } } dx$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9 Q20.1

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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