If \(\int f(x) d x=g(x)+c\), then \(\int f(x) g^{\prime}(x) d x\)

(a)

\(\int (f(x))^2dx\)

(b)

\(\int f(x)g(x)dx\)

(c)

\(\int f'(x)g(x)dx\)

(d)

\(\int (g(x))^2dx\)

If \(\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x\) is

(a)

cot(xe^x)+c

(b)

sec(xe^x)+c

(c)

tan(xe^x)+c

(d)

cos(xe^x)+c

\(\int \sin ^3 x d x\) is

(a)

\({-3\over 4}cos \ x-{cos \ 3x\over 12}+c\)

(b)

\({3\over 4}cos \ x+{cos \ 3x\over 12}+c\)

(c)

\({-3\over 4}cos \ x+{cos \ 3x\over 12}+c\)

(d)

\({-3\over 4}sin \ x-{sin \ 3x\over 12}+c\)

\(\int \frac{e^x\left(x^2 \tan ^{-1} x+\tan ^{-1} x+1\right)}{x^2+1} d x\) is

(a)

e^x tan^-1(x+1)+c

(b)

tan^-1(e^x)+c

(c)

\(e^x{(tan^{-1}x)^2\over 2}+c\)

(d)

e^x tan^-1 x+c

\(\int \frac{\sec ^2 x}{\tan ^2 x-1} d x\) is

(a)

\(2log|{1-tan x\over 1+tan \ x}|+c\)

(b)

\(log|{1+tan x\over 1-tan \ x}|+c\)

(c)

\({1\over2}log|{tan x+1\over tan \ x-1}|+c\)

(d)

\({1\over2}log|{tan x-1\over tan \ x+1}|+c\)

\(\int e^{\sqrt{x}} d x\) is

(a)

\(2\sqrt{x}(1-e^{\sqrt{x}})+c\)

(b)

\(2\sqrt{x}(e^{\sqrt{x}}-1)+c\)

(c)

\(2e^{\sqrt{x}}(1-\sqrt{x})+c\)

(d)

\(2e^{\sqrt{x}}(\sqrt{x}-1)+c\)

\(\int { sin } \)e^x . d (e^x) = ________+c.

(a)

cos (e^x)

(b)

sin(e^x)

(c)

- cos (e^x)

(d)

-sin(e^x)

\(\int { \frac { 4\left( sin^{ -1 }x \right) ^{ 3 } }{ \sqrt { 1-{ x }^{ 2 } } } } \) dx = ________+c.

(a)

log (sin ^-1x)

(b)

(sin ^-1 x)⁴

(c)

4 (sin^-1 x)⁴

(d)

\(\frac { \left( { sin }^{ -1 }x \right) ^{ 4 } }{ 4 } \)

\(\int { \frac { { e }^{ x } }{ \left( 1+{ e }^{ x } \right) ^{ 2 } } } \) dx =_______+c

(a)

\(\frac { 1 }{ 1+{ e }^{ x } } \)

(b)

-\(\frac { 1 }{ 1+{ e }^{ x } } \)

(c)

1 + e^x

(d)

\(\frac { \left( 1+{ e }^{ x } \right) ^{ 3 } }{ 3 } \)

\(\int { { tan }^{ 3 } } 2sec2x\) dx = ___________+c.

(a)

\(\frac { 1 }{ 6 } \) sec³ 2x

(b)

\(\frac { 1 }{ 6 } \) sec³2x - \(\frac { 1 }{ 2 } \) sec 2x

(c)

\(\frac { 1 }{ 2 } \) sec 2x

(d)

\(\frac { 1 }{ 6 } \) sec³ 2x + \(\frac { 1 }{ 2 } \) sec 2x