Question 1: Palette 1/7

JEE Advanced 2012 MCQ

The integral $\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x$ equals (for some arbitrary constant $K$)

A

$-\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^2\right\}+K$

B

$\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^2\right\}+K$

C

$-\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K$

D

$\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K$

Question 2: Palette 2/7

JEE Advanced 2008 MCQ

Let $I = \int {{{{e^x}} \over {{e^{4x}} + {e^{2x}} + 1}}dx,\,\,J = \int {{{{e^{ - x}}} \over {{e^{ - 4x}} + {e^{ - 2x}} + 1}}dx.} } $ Then

for an arbitrary constant $C$, the value of $J -I$ equals :

A

${1 \over 2}\log \left( {{{{e^{4x}} - {e^{2x}} + 1} \over {{e^{4x}} + {e^{2x}} + 1}}} \right) + C$

B

${1 \over 2}\log \left( {{{{e^{2x}} + {e^x} + 1} \over {{e^{2x}} - {e^x} + 1}}} \right) + C$

C

${1 \over 2}\log \left( {{{{e^{2x}} - {e^x} + 1} \over {{e^{2x}} + {e^x} + 1}}} \right) + C$

D

${1 \over 2}\log \left( {{{{e^{4x}} + {e^{2x}} + 1} \over {{e^{4x}} - {e^{2x}} + 1}}} \right) + C$

Question 3: Palette 3/7

JEE Advanced 2007 MCQ

Let $F(x)$ be an indefinite integral of $si{n^2}x.$

STATEMENT-1: The function $F(x)$ satisfies $F\left( {x + \pi } \right) = F\left( x \right)$
for all real $x$. because

STATEMENT-2: ${\sin ^2}\left( {x + \pi } \right) = {\sin ^2}x$ for all real $x$.

A

Statement-1 is True, Statement-2 is True; Statement-2 is is a correct explanation for Statement-1.

B

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

C

Statement- 1 is True, Statement-2 is False.

D

Statement-1 is False, Statement-2 is True.

Question 4: Palette 4/7

JEE Advanced 2007 MCQ

Let $f(x)=\frac{x}{\left(1+x^{n}\right)^{1 / n}}$ for $n \geq 2$ and $g(x)=\underbrace{(f o f o \ldots . o f)}_{f \text { occurs } n \text { times }}(x)$. Then $\int x^{n-2} g(x) d x$ equals :

A

$\frac{1}{n(n-1)}\left(1+n x^{n}\right)^{1-\frac{1}{n}}+k$

B

$\frac{1}{n(n+1)}\left(1+n x^{n}\right)^{1-\frac{1}{n}}+k$

C

$\frac{1}{n(n-1)}\left(1+n x^{n}\right)^{1-\frac{1}{n}}+k$

D

$\frac{1}{(n+1)}\left(1+n x^{n}\right)^{1+\frac{1}{n}}+k$

Question 5: Palette 5/7

JEE Advanced 2006 MCQ

$\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x$ is equal to

A

$\frac{\sqrt{2 x^4-2 x^2+1}}{x^2}+\mathrm{C}$

B

$\frac{\sqrt{2 x^4-2 x^2+1}}{x^3}+\mathrm{C}$

C

$\frac{\sqrt{2 x^4-2 x^2+1}}{x}+\mathrm{C}$

D

$\frac{\sqrt{2 x^4-2 x^2+1}}{2 x^2}+C$

Question 6: Palette 6/7

JEE Advanced 2005 MCQ

If $\int\limits_{\sin x}^1 {{t^2}f\left( t \right)dt = 1 - \sin x,} $ then f$\left( {{1 \over {\sqrt 3 }}} \right)$ is

A

${1 \over 3}$

B

${{1 \over {\sqrt 3 }}}$

C

$3$

D

${\sqrt 3 }$

Question 7: Palette 7/7

JEE Advanced 1995 MCQ

The value of the integral $\int {{{{{\cos }^3}x + {{\cos }^5}x} \over {{{\sin }^2}x + {{\sin }^4}x}}} \,dx\,$ is

A

$\sin x - 6{\tan ^{ - 1}}\left( {\sin x} \right) + c$

B

$\sin x - 2{\left( {\sin x} \right)^{ - 1}} + c$

C

$\sin x - 2{\left( {\sin x} \right)^{ - 1}} - 6{\tan ^{ - 1}}\left( {\sin x} \right) + c$

D

$\,\sin x - 2{\left( {\sin x} \right)^{ - 1}} + 5{\tan ^{ - 1}}\left( {\sin x} \right) + c$