If \(\int {3^{1\over x}\over x^2}dx=k(3^{1\over x})+c\), then the value of k is

(a)

log 3

(b)

-log 3

(c)

\(-{1\over log3}\)

(d)

\({1\over log3}\)

\(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} d x\) is

(a)

x+c

(b)

\({x^3\over 3}+c\)

(c)

\({3\over x^3}+c\)

(d)

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

\(\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x\) is

(a)

\({1\over2}sin 2x+c\)

(b)

\(-{1\over2}sin 2x+c\)

(c)

\({1\over2}cos 2x+c\)

(d)

\(-{1\over2}cos 2x+c\)

\(\int \frac{x^2+\cos ^2 x}{x^2+1} \operatorname{cosec}^2 x d x\) is

(a)

cot x + sin ^-1x + c

(b)

-cot x + tan^-1x + c

(c)

-tan x + cot^-1x + c

(d)

-cot x - tan^-1x + c

\(\int e^{-4 x} \cos x d x\) is

(a)

\({e^{-4x}\over 17}[4cos \ x-sin \ x]+c\)

(b)

\({e^{-4x}\over 17}[-4cos \ x+sin \ x]+c\)

(c)

\({e^{-4x}\over 17}[4cos \ x+sin \ x]+c\)

(d)

\({e^{-4x}\over 17}[-4cos \ x-sin \ x]+c\)

\(\int \frac{x+2}{\sqrt{x^2-1}} d x\) is

(a)

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

(b)

\(sin^{-1}x-2 log|x+\sqrt{x^2-1}|+c\)

(c)

\(2 log|x+\sqrt{x^2-1}|-sin^{-1}x+c\)

(d)

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

\(\int \sin \sqrt{x} d x\) is

(a)

\(2(-\sqrt{x}cos\sqrt{x}+sin\sqrt{x})+c\)

(b)

\(2(-\sqrt{x}cos\sqrt{x}-sin\sqrt{x})+c\)

(c)

\(2(-\sqrt{x}sin\sqrt{x}-cos\sqrt{x})+c\)

(d)

\(2(-\sqrt{x}sin\sqrt{x}+cos\sqrt{x})+c\)

\(\int { \frac { \left( 1+logx \right) ^{ 2 } }{ x } } \) dx = _______+C.

(a)

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

(b)

3 log ( 1 + log x)

(c)

2(1 + log x)

(d)

none of these

\(\int { \frac { 1 }{ 9x^{ 2 }-4 } } \) dx = ________+c.

(a)

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

(b)

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

(c)

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

(d)

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

\(\int { \frac { x }{ 4+{ x }^{ 4 } } } \) dx is equal to________+c.

(a)

\(\frac { 1 }{ 4 } \) tan^-1 (x²)

(b)

\(\frac { 1 }{ 4 } \) tan^-1 \(\left( \frac { { x }^{ 2 } }{ 2 } \right) \)

(c)

\(\frac { 1 }{ 2 } \) tan^-1 \(\left( \frac { { x }^{ 2 } }{ 2 } \right) \)

(d)

none of these