The integral $\int {\sec }^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right){\csc }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $- 3 {\cot }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) + \mathrm{C}$

• $- 3 {\tan }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

• $3 {\tan }^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

• $- \frac{3}{4} {\tan }^{\raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$

If $\int \frac{\mathrm{dx}}{{\left({\mathrm{x}}^2 - 2\mathrm{x} + 10\right)}^2} = \mathrm{A}\left({\tan }^{-1}\left(\frac{\mathrm{x} - 1}{3}\right) + \frac{\mathrm{f}\left(\mathrm{x}\right)}{{\mathrm{x}}^2 - 2\mathrm{x} + 10}\right) + \mathrm{C}$ where C is a constant of integration, then

• $\mathrm{A} = \frac{1}{54}, \mathrm{f}\left(\mathrm{x}\right) = 3\left(\mathrm{x} - 1\right)$

• $\mathrm{A} = \frac{1}{54}, \mathrm{f}\left(\mathrm{x}\right) = 9{\left(\mathrm{x} - 1\right)}^2$

• $\mathrm{A} = \frac{1}{27}, \mathrm{f}\left(\mathrm{x}\right) = 9\left(\mathrm{x} - 1\right)$

• $\mathrm{A} = \frac{1}{81}, \mathrm{f}\left(\mathrm{x}\right) = 3\left(\mathrm{x} - 1\right)$

323.

The value of ${\int }_0^{2\mathrm{\pi }}\left[\sin \left(2\mathrm{x}\right)\left(1 + \cos \left(3\mathrm{x}\right)\right)\right]\mathrm{dx}$, (where [t] denotes Greatest Integer Function)

• $- 2\mathrm{\pi }$

• $\mathrm{\pi }$

• $2\mathrm{\pi }$

• $- \mathrm{\pi }$

If $\int {\mathrm{x}}^5{\mathrm{e}}^{- {\mathrm{x}}^2}\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right){\mathrm{e}}^{- 2} + \mathrm{c}$, where c is a constant of interation then g(- 1) is equal to :

• - 1

• 1

• $\frac{1}{2}$

• $- \frac{5}{2}$

The integral ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\sec }^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left({\csc }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(\mathrm{x}\right)\right)\mathrm{dx}$ =

• 3^5/3 - 3^1/3

• 3^7/6 - 3^5/6

• 3^5/6 - 3^1/3

• 3^4/3 - 3^1/3

Primitive of cos^-1(x) w.r.t. x is

• ${\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) - \frac{1}{2}\sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

• ${\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) - \sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

• ${\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) + \sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

• ${\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) + \frac{1}{2}\sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

If y = log(cot(x)), then ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{ydx}$ is equal to

• 1

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

$\int \frac{1}{{\sin }^2\left(\mathrm{x}\right) + {\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + \mathrm{c}$

• $\tan \left(\mathrm{x}\right) + \cot \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\tan \left(\mathrm{x}\right) - \cot \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

   

Primitive of $\frac{1}{4\sqrt{\mathrm{x}} + \mathrm{x}}$ is equal to

• $2\log \left|1 + 4\sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{2}\log \left|4 - \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $2\log \left| 4 + \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{2}\log \left| 4 + \sqrt{\mathrm{x}}\right| +\hspace{0.17em}\mathrm{c}$

$\int {\mathrm{e}}^{\mathrm{x}}\left(\log \left(\mathrm{x}\right) + \frac{1}{\mathrm{x}}\right)\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{\mathrm{x}}\log \left(\mathrm{x}\right) + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\log \left(\mathrm{x}\right)}+ \mathrm{c}$

• $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} +\hspace{0.17em}\mathrm{c}$