Question 2 :
$\int { \cfrac { { x }^{ 2 }+1 }{ x({ x }^{ 2 }-1) } dx } $ is equal to 

Question 4 :
. Let $\displaystyle \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{(1+\mathrm{x}^{\mathrm{n}})^{1/\mathrm{n}}}$ for $\mathrm{n}\geq 2$ and $\displaystyle \mathrm{g}(\mathrm{x})=\frac{(\mathrm{f}\mathrm{o}\mathrm{f}\mathrm{o}\ldots \mathrm{o}\mathrm{f})}{\mathrm{f}\mathrm{o}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{s}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}(\mathrm{x})$ . Then $\displaystyle \int \mathrm{x}^{\mathrm{n}-2}\mathrm{g}$ (x)dx equals&nbsp;<br><br>

Question 5 :
If $f\left(\displaystyle\frac{3x-4}{3x+4}\right)=x+2, x\neq -\displaystyle\frac{4}{3}$, and $\displaystyle\int f(x)dx=A\log |1-x|+Bx+C$, then the ordered pair $(A, B)$ is equal to (where C is a constant of integration)

Question 8 :
If&nbsp;$\displaystyle I = \int (x^{1/3} + (tan \: x)^{1/3}) dx = \frac {A}{512} \log \frac {t^4 = t^2 + 1}{(t^2 + 1)^2} + \frac {\sqrt 3}{2} tan^{-1} \left ( \frac {2t^2 - 1}{2 \sqrt 3} \right ) + \frac {3}{4} (tan^{-1} t^3)^4 + C$ where&nbsp;$\displaystyle t^3 = tan \: x$ then A is equal to

Question 11 :
$\displaystyle \int \dfrac {1}{x^{2}(x^{4} + 1)^{3/4}} dx$ is equal to ____

Question 13 :
If $I =\displaystyle \int {\dfrac{{dx}}{{{{\left( {2ax + {x^2}} \right)}^{\frac{3}{2}}}}}} $, then $I$ is equal to&nbsp;

Question 14 :
If&nbsp;$\displaystyle I = \int tan^{-1} \sqrt {\left ( \sqrt x - 1 \right )} dx = (u^2 + 1)^2 tan^{-1} u - \frac {A}{1863} u^3 - u + C$ where&nbsp;$\displaystyle u = \sqrt {\sqrt x - 1}$ then A is equal to.

Question 15 :
The value of the expression $\dfrac{\int_{0}^{a} x^{4} \sqrt{a^{2}-x^{2}} d x}{\int_{0}^{a} x^{2} \sqrt{a^{2}-x^{2}} d x}$ is equal to

Question 19 :
Solution of the differential equation<br>$\left \{\dfrac {1}{x} - \dfrac {y^{2}}{(x - y)^{2}}\right \} dx + \left \{\dfrac {x^{2}}{(x - y)^{2}} - \dfrac {1}{y}\right \} dy = 0$ is<br>(where $c$ is arbitrary constant).

Question 21 :
If $\int { f\left( x \right) dx=\Psi \left( x \right)&nbsp; }$, then $\int { { x }^{ 5 }f\left( { x }^{ 3 } \right)&nbsp; } dx$ is equal to:

Question 22 :
Integrate&#160;<br/>$\displaystyle\int {\dfrac{{dx}}{{\left( {x + 1} \right)\sqrt {2{x^2} + 3x + 1} }}} $

Question 27 :
$\displaystyle \int \sqrt {1+x \sqrt {1+(x+1) \sqrt {1+(x+2) (x+4)}}}$ $dx$ is equal to

Question 28 :
Let $F(x)$ be the primitive of $\displaystyle\frac{3x+2}{\sqrt{x-9}}$ with respect to $x$. If $F(10)=60$, then the value of $F(13)$ is equal to

Question 29 :
If $M= \displaystyle \int _{ 0 }^{ \pi /2 }{ \cfrac { \cos { x }&#160; }{ x+2 }&#160; } dx,N=\int _{ 0 }^{ \pi /4 }{ \cfrac { \sin { x } \cos { x }&#160; }{ { \left( x+1 \right)&#160; }^{ 2 } }&#160; } dx\quad $, then the value of $M-N$ is ?

Question 31 :
$\displaystyle&#160;\int { \frac { 1+x }{ 1+\sqrt [ 3 ]{ x } &#160;} dx } $ is equal to

Question 33 :
If $ \displaystyle f(x)=\lim_{n\rightarrow \infty }(2x+4x^{3}+......+2^{n}x^{2n-1})\left ( 0&lt;x&lt;\frac{1}{\sqrt{2}} \right )$, then the value of $\displaystyle\int f(x) dx$ is equal to<br/><div>$\textbf{Note}$: $c$ is the constant of integration.</div>

Question 34 :
If $f\left( \cfrac { 3x-4 }{ 3x+4 }&nbsp; \right) =x+2$, then $\int { f(x) } dx$ is

Question 35 :
$\displaystyle\int { \dfrac { x+2 }{ \left( { x }^{ 2 }+3x+3 \right) \sqrt { x+1 } &nbsp;} dx } $ is equal to

Question 38 :
If $\displaystyle{\int \frac{\displaystyle dx}{\displaystyle&#160;\sqrt{x}+\displaystyle \sqrt[3]{x}}}=a\sqrt{x}+b(\sqrt[3]{x})+c(\sqrt[6]{x})+d\: \ln(\sqrt[6]{x}+1)+e$, $e$ being arbitrary constant then. Find the value of $20a + b + c + d.$<br/>

Question 44 :
$\displaystyle \int &nbsp;\frac{x\quad dx}{\sqrt{1 + x^{2} + \sqrt{(1 + x^{2})^{3}}}}$ is equal to

Question 46 :
$\displaystyle\int { \dfrac { 1 }{ { x }^{ 2 }{ \left( { x }^{ 4 }+1 \right) &nbsp;}^{ { 3 }/{ 4 } } } dx } $ is equal to

Question 48 :
Evaluate: $\displaystyle \int _{ 0 }^{ \tfrac { \pi &#160;}{ 4 } &#160;}{ \cfrac { \sin { x } +\cos { x } &#160;}{ 7+9\sin { 2x } &#160;} &#160;} dx$

Question 51 :
Evaluate: $\displaystyle \int _{ 0 }^{ \tfrac{\pi}4 }{ \left[ \sqrt { \tan { x } &#160;} +\sqrt { \cot { x } &#160;} &#160;\right] &#160;} dx$

Question 52 :
Integrate the given equation $\int { \dfrac { \sqrt { { x }^{ 2 }+1 } (\log { \left( { x }^{ 2 }+1 \right) -2\log { x })&#160; }&#160; }{ { x }^{ 4 } }&#160; } dx$