NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

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NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.6

Question 1.
x sinx
Solution:
By part integration
∫x sinx dx = x(-cosx) – ∫1(-cosx)dx
= – x cosx + ∫cosxdx
= – x cosx + sinx + c

Question 2.
x sin 3x
Solution:
∫xsin 3x dx = x ∫sin 3 dx – ∫(\(\frac { d }{ dx }\)(x) ∫sin 3x dx)dx
= \(=x\left(\frac{-\cos 3 x}{3}\right)-\int 1\left(\frac{-\cos 3 x}{3}\right) d x\)
= \(\frac{-1}{3} x \cos 3 x+\frac{1}{9} \sin 3 x+C\)

Question 3.
\({ x }^{ 2 }{ e }^{ x }\)
Solution:
\(\int { { x }^{ 2 }{ e }^{ x } } dx={ x }^{ 2 }{ e }^{ x }-2{ x }{ e }^{ x }+2{ e }^{ x }+c\)
\(={ e }^{ x }\left( { x }^{ 2 }-2x+2 \right) +c\)

Question 4.
x logx
Solution:

Question 5.
x log 2x
Solution:

Question 6.
\({ x }^{ 2 }logx\)
Solution:

Question 7.
x sin^-1 x
Solution:

Question 8.
x tan^-1 x
Solution:
\(I=x\quad { tan}^{ -1 }x.\left( \frac { { x }^{ 2 } }{ 2 } \right) -\int { \frac { 1 }{ \sqrt { 1+{ x }^{ 2 } } } } .\frac { { x }^{ 2 } }{ 2 } dx\)
\(=\frac { { x }^{ 2 } }{ 2 } { tan }^{ -1 }x-\frac { 1 }{ 2 } \int { \left( 1-\frac { 1 }{ 1+{ x }^{ 2 } } \right) dx } \)
\(=\frac { { x }^{ 2 } }{ 2 } { tan }^{ -1 }x-\frac { 1 }{ 2 } x+\frac { 1 }{ 2 } { tan }^{ -1 }\)

Question 9.
x cos^-1 x
Solution:

Question 10.
(sin^-1 x)²
Solution:
Method I
Take 1 as the second function and integrate, w.r.t. x

Method II

Method III

Question 11.
\(\frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}}\)
Solution:

Question 12.
x sec²x
Solution:
∫x sec²x dx
= \(x \int \sec ^{2} x d x-\int\left(\frac{d}{d x}(x) \int \sec ^{2} x d x\right) d x\)
= x tan x -∫tan x dx
= x tan x – log |sec x| + C
= x tan x + log |cos x| + C

Question 13.
tan^-1 x
Solution:
\(\int { { tan }^{ -1 }xdx } =x{ tan }^{ -1 }x-\frac { 1 }{ 2 } \int { \frac { 2x }{ 1+{ x }^{ 2 } } dx } \)
\(=x{ tan }^{ -1 }x-\frac { 1 }{ 2 } log|1+{ x }^{ 2 }|+C \)

Question 14.
x(logx)²
Solution:

Question 15.
(x² + 1) log x
Solution:

Question 16.
\({ e }^{ x }(sinx+cosx)\)
Solution:

Question 17.
\(\frac { { xe }^{ x } }{ { (1+x) }^{ 2 } } \)
Solution:

Question 18.
\(\frac { { e }^{ x }(1+sinx) }{ 1+cosx } \)
Solution:

Question 19.
\({ e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ { x }^{ 2 } } \right) \)
Solution:

Question 20.
\(\frac { { (x-2)e }^{ x } }{ { (x-1) }^{ 3 } } \)
Solution:

Question 21.
\({ e }^{ 2x }sinx\)
Solution:

Question 22.
\(\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)\)
Solution:

Question 23.
\(\int { { x }^{ 2 }{ e }^{ { x }^{ 3 } } } dx\quad equals\)
(a) \(\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+C\)
(b) \(\frac { 1 }{ 3 } +{ e }^{ { x }^{ 2 } }+C\)
(c) \(\frac { 1 }{ 2 } { e }^{ { x }^{ 3 } }+C\)
(d) \(\frac { 1 }{ 2 } { e }^{ { x }^{ 2 } }+C\)
Solution:
(a) \(\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+C\)

Question 24.
\(\int { { e }^{ x }secx(1+tanx) } dx\quad equals\)
(a) \({ e }^{ x }cosx+C\)
(b) \({ e }^{ x }secx+C\)
(c) \({ e }^{ x }sinx+C\)
(d) \({ e }^{ x }tanx+C\)
Solution:
(b) \({ e }^{ x }secx+c\)