\(\int { \left( x-1 \right) } { e }^{ -x }\) dx = __________ +c

(a)

-xe^x

(b)

xe^x

(c)

-xe^-x

(d)

xe^-x

If \(\int { \frac { { 2 }^{ \frac { 1 }{ x } } }{ { x }^{ 2 } } } dx=k{ 2 }^{ \frac { 1 }{ x } }\) +c, then k is ___________

(a)

\(-\frac { 1 }{ { log }_{ e }2 } \)

(b)

- log_e2

(c)

-1

(d)

\(\frac { 1 }{ 2 } \)

\(\int { { \left| x \right| }^{ 3 } } \)dx = ________________ +c

(a)

\(\frac { { -x }^{ 4 } }{ 4 } \)

(b)

\(\frac { { \left| x \right| }^{ 4 } }{ 4 } \)

(c)

\(\frac { { x }^{ 4 } }{ 4 } \)

(d)

none of these

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

(a)

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

(b)

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

(c)

\(\frac { 1 }{ { \left( { e }^{ x }+1 \right) }^{ 2 } } \)

(d)

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

\(\int { { e }^{ x } } \) (1-cot x +cot² x) dx = _______________ +c

(a)

e^x cot x

(b)

- e^x cot x

(c)

e^x cosec x

(d)

-e^x cosec x

\(\int { { e }^{ x } } \) f(x) + f' (x) dx = _____________ +c

(a)

e^x f(x)

(b)

e^x + f(x)

(c)

2e^x f(x)

(d)

e^x - f(x)

If ∫ x sin x dx = - x cos x + α then α = __________ +c

(a)

sin x

(b)

cos x

(c)

C

(d)

none of these

If \(\int { \frac { 1 }{ \left( x+2 \right) \left( { x }^{ 2 }+1 \right) } } \) dx = a log \(\left| 1+{ x }^{ 2 } \right| \) +b tan^-1 x + \(\frac { 1 }{ 5 } log\left| x+2 \right| \) +c then ___________

(a)

\(a=-\frac { 1 }{ 10 } ,b=\frac { -2 }{ 5 } \)

(b)

\(a=\frac { 1 }{ 10 } ,b=\frac { -2 }{ 5 } \)

(c)

\(a=-\frac { 1 }{ 10 } ,b=\frac { 2 }{ 5 } \)

(d)

\(a=\frac { 1 }{ 10 } ,b=\frac { 2 }{ 5 } \)

\(\int { { 3 }^{ x+2 } } \) dx = ______________ +c

(a)

\(\frac { { 3 }^{ x } }{ log3 } \)

(b)

\(\frac { 9\left( { 3 }^{ x } \right) }{ log3 } \)

(c)

\(\frac { 3.{ 3 }^{ x } }{ log3 } \)

(d)

\(\frac { { 3 }^{ x } }{ 9log3 } \)

\(\int { \left( \frac { x }{ m } +\frac { m }{ x } \right) } \) dx = __________ +c

(a)

\(\frac { { x }^{ 2 } }{ 2m } +m\log { \left| x \right| } \)

(b)

\(\frac { x }{ { m }^{ 2 } } +m\log { \left| x \right| } \)

(c)

\(-\frac { 1 }{ { mx }^{ 2 } } +m\log { \left| x \right| } \)

(d)

\(\frac { 1 }{ m } -\frac { m }{ { x }^{ 2 } } \)

The anti-derivative of f(x) = \(\sqrt { x } +\frac { 1 }{ \sqrt { x } } \) is ___________ +c

(a)

\(\frac { { 2 } }{ 3 } { x }^{ \frac { 3 }{ 2 } }+\frac { 2 }{ { x }^{ \frac { 1 }{ 2 } } } \)

(b)

\(\frac { { 3 } }{ 2 } { x }^{ \frac { 3 }{ 2 } }+2{ x }^{ \frac { 1 }{ 2 } }\)

(c)

\(\frac { { 2 } }{ 3 } { x }^{ \frac { 3 }{ 2 } }+2{ x }^{ \frac { 1 }{ 2 } }\)

(d)

none

\(\int { { a }^{ 3x+2 } } \) dx = _____________ +c

(a)

a^3x+2

(b)

\(\frac { { a }^{ 3x+2 } }{ 3 } \)

(c)

\(\frac { { a }^{ 3x+2 } }{ 3loga } \)

(d)

3 log a (a^3x+2)

∫ e^{3 log x} (x⁴ +1)^-1 dx = ____________ +c

(a)

\(\log { \left| { x }^{ 4 }+1 \right| } \)

(b)

\(4\log { \left| { x }^{ 4 }+1 \right| } \)

(c)

-4 log |x⁴ +1|

(d)

\(\frac { 1 }{ 4 } \log { \left| { x }^{ 4 }+1 \right| } \)

\(\int { \frac { dx }{ 9{ x }^{ 2 }-4 } } \)= ____________ +c

(a)

\(\log { \left| \frac { 3x-2 }{ 3x+2 } \right| } \)

(b)

\(12\log { \left| \frac { 3x-2 }{ 3x+2 } \right| } \)

(c)

\(\frac { 1 }{ 12 } \log { \left| \frac { 3x-2 }{ 3x+2 } \right| } \)

(d)

\(\frac { 1 }{ 6 } \log { \left| \frac { 3x-2 }{ 3x+2 } \right| } \)

\(\int _{ 1 }^{ e }{ log } x\) dx = __________ +c

(a)

1

(b)

e-1

(c)

e+1

(d)

0