These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 Questions and Answers are prepared by our highly skilled subject experts.
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.2
Question 1.
\(\frac { 2x }{ 1+{ x }^{ 2 } }\)
Solution:
Let 1 + x² = t
⇒ 2xdx = dt
∴ ∫\(\frac { 2x }{ 1+{ x }^{ 2 } }\) dx
= ∫\(\frac { dt }{ t }\)
= log t + C
= log(1 + x²) + C
Question 2.
\(\frac { { \left( logx \right) }^{ 2 } }{ x } \)
Solution:
Let logx = t
⇒ \(\frac { 1 }{ x }dx=dt\)
\(\int \frac{(\log x)^{2}}{x} d x=\int t^{2} d t=\frac{t^{3}}{3}+\mathrm{C}\)
= \(\frac{1}{3}(\log x)^{3}+\mathrm{C}\)
Question 3.
\(\frac { 1 }{ x+xlogx }\)
Solution:
Put 1 + logx = t
∴ \(\frac { 1 }{ x }\)dx = dt
\(\int { \frac { 1 }{ x(1+logx) } dx } =\int { \frac { 1 }{ t } dt } =log|t| + C \)
= log|1+logx| + C
Question 4.
sinx sin(cosx)
Solution:
∫sinx sin(cosx)dx = ∫sin t dt
= – (-cos t) + C
= – cost + C
= cos(cos x) + C
Question 5.
sin(ax+b) cos(ax+b)
Solution:
Another method:
Question 6.
\(\sqrt { ax+b }\)
Solution:
Question 7.
x\(\sqrt { x+2 }\)
Solution:
Question 8.
x\(\sqrt { 1+{ 2x }^{ 2 } }\)
Solution:
Question 9.
(4x+2)\(\sqrt { { x }^{ 2 }+ x + 1 } \)
Solution:
Question 10.
\(\frac { 1 }{ x-\sqrt { x } } \)
Solution:
Question 11.
\(\frac { x }{ \sqrt { x+4 } }\) , x > 0
Solution:
Question 12.
\({ { (x }^{ 3 }-1) }^{ \frac { 1 }{ 3 } }.{ x }^{ 5 }\)
Solution:
Question 13.
\(\frac { { x }^{ 2 } }{ { { (2+3x }^{ 3 }) }^{ 3 } } \)
Solution:
Question 14.
\(\frac { 1 }{ x(logx)^{ m } }\), x > 0
Solution:
Put log x = t, \(\frac { dt }{ dx }\) = \(\frac { 1 }{ x }\) ⇒ dt = \(\frac { 1 }{ x }\)dx
\(\int \frac{1}{x(\log x)^{m}} d x=\int \frac{d t}{t^{m}}=\int t^{-m} d t=\frac{t^{-m+1}}{-m+1}+\mathrm{C}\)
\(=\frac { { (logx) }^{ 1-m } }{ 1-m }\) + C
Question 15.
\(\frac { x }{ 9-4{ x }^{ 2 } } \)
Solution:
Question 16.
\({ e }^{ 2x+3 }\)
Solution:
put 2x + 3 = t
so that 2dx = dt
\(\int { { e }^{ 2x+3 } } dx\quad =\frac { 1 }{ 2 } \int { { e }^{ t }dt } \quad =\frac { 1 }{ 2 } { e }^{ t }+c\quad =\frac { 1 }{ 2 } { e }^{ 2x+3 }\) + C
Question 17.
\(\frac { x }{ { e }^{ { x }^{ 2 } } } \)
Solution:
Question 18.
\(\frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } } \)
Solution:
let \(\quad { tan }^{ -1 }x=t\Rightarrow \frac { 1 }{ 1+{ x }^{ 2 } } dx=dt\)
∴ \(\frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } } \)dx
= ∫et dt
= et+C
= etan-1 + C
Question 19.
\(\frac { { e }^{ 2x }-1 }{ { e }^{ 2x }+1 } \)
Solution:
Another method
Question 20.
\(\frac { { e }^{ 2x }-{ e }^{ 2x } }{ { e }^{ 2x }+{ e }^{ -2x } } \)
Solution:
Question 21.
tan²(2x – 3)
Solution:
∫tan²(2x – 3)dx = ∫[sec²(2x – 3) – 1]dx = I
put 2x – 3 = t
so that 2dx = dt
I = \(\frac { 1 }{ 2 }\) ∫sec²t dt – x + C
= \(\frac { 1 }{ 2 }\)t – x + C
= \(\frac { 1 }{ 2 }\)tan(2x -3 ) – x + C
Question 22.
sec²(7 – 4x)
Solution:
∫sec²(7 – 4x)dx
= \(\frac { tan(7-4x) }{ -4 }\) + C
Question 23.
\(\frac { { sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } \)
Solution:
Let \(\quad { sin }^{ -1 }x=t\quad\Rightarrow\frac { 1dx }{ \sqrt { 1-{ x }^{ 2 } } }\) = dt
\(\int { \frac { { sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } dx } =\int { t\quad dt } =\frac { 1 }{ 2 } { t }^{ 2 }+c=\frac { 1 }{ 2 } { { (sin }^{ -1 }x) }^{ 2 }+c\)
Question 24.
\(\frac { 2cosx-3sinx }{ 6cosx+4sinx }\)
Solution:
Question 25.
\(\frac { 1 }{ { cos }^{ 2 }x{ (1-tanx) }^{ 2 } } \)
Solution:
Question 26.
\(\frac { cos\sqrt { x } }{ \sqrt { x } } \)
Solution:
Question 27.
\(\sqrt { sin2x } cos2x\)
Solution:
Question 28.
\(\frac { cosx }{ \sqrt { 1+sinx } } \)
Solution:
Question 29.
cotx log sinx
Solution:
Question 30.
\(\frac { sinx }{ 1+cosx }\)
Solution:
put 1 + cosx = t
⇒ – sinx dx = dt
∫\(\frac { sinx }{ 1+cosx }\)dx
= – \(\frac { dt }{ t }\)
= – log|t| + c
= – log|1 + cosx| + C
= \(\log \left|\frac{1}{1+\cos x}\right|+\mathrm{C}\)
Question 31.
\(\frac { sinx }{ { (1+cosx) }^{ 2 } } \)
Solution:
put 1 + cosx = t
so that – sinx dx = dt
∫\(\frac { sinx }{ { (1+cosx) }^{ 2 } } \) = – \(\int \frac{d t}{t^{2}}=\frac{1}{t}+C\)
= \(\frac { 1 }{ t }\) + C
= \(\frac { 1 }{ 1 + cosx }\) + C
Question 32.
\(\frac { 1 }{ 1 + cotx }\)
Solution:
Question 33.
\(\frac { 1 }{ 1-tanx }\)
Solution:
Question 34.
\(\frac { \sqrt { tanx } }{ sinxcosx } \)
Solution:
Question 35.
\(\frac { { (1+logx) }^{ 2 } }{ x } \)
Solution:
Question 36.
\(\frac { (x+1){ (x+logx) }^{ 2 } }{ x } \)
Solution:
Question 37.
\(\frac { { x }^{ 3 }sin({ tan }^{ -1 }{ x }^{ 4 }) }{ 1+{ x }^{ 8 } } dx \)
Solution:
Question 38.
\(\int { \frac { { 10x }^{ 9 }+{ 10 }^{ x }log{ e }^{ 10 } }{ { x }^{ 10 }+{ 10 }^{ x } } dx } \) equals
(a) 10x – x10 + C
(b) 10x + x10 + C
(c) (10x – x10) + C
(d) log (10x + x10) + C
Solution:
(d) log (10x + x10) + C
Question 39.
\(\int { \frac { dx }{ { sin }^{ 2 }x{ \quad cos }^{ 2 }x } = } \)
(a) tan x + cot x + c
(b) tan x – cot x + c
(c) tan x cot x + c
(d) tan x – cot 2 x + c
Solution:
(b) tan x – cot x + c
