Which of the following is true ?

• ${\int }_0^1{\mathrm{e}}^{\mathrm{x}}\mathrm{dx} = \mathrm{e}$

• ${\int }_0^1{2}^{\mathrm{x}}\mathrm{dx} = \log \left(2\right)$

• ${\int }_0^1\sqrt{\mathrm{x}}\mathrm{dx} = \frac{2}{3}$

• ${\int }_0^1\mathrm{xdx} = \frac{1}{3}$

$\int \left[\sin \left(\log \left(\mathrm{x}\right)\right) + \cos \left(\log \left(\mathrm{x}\right)\right)\right]\mathrm{dx}$ is equal to

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

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

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

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

$\int {\mathrm{e}}^{\mathrm{x}}\frac{\left(\mathrm{x} - 1\right)}{{\mathrm{x}}^2}\mathrm{dx}$ is equal to

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^2} + \mathrm{c}$

• $\frac{- {\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^2} + \mathrm{c}$

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

• $\frac{- {\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{c}$

${\int }_5^{10}\frac{1}{\left(\mathrm{x} - 1\right)\left(\mathrm{x} - 2\right)}\mathrm{dx}$

• $\log \left(\frac{27}{32}\right)$

• $\log \left(\frac{32}{27}\right)$

• $\log \left(\frac{8}{9}\right)$

• $\log \left(\frac{3}{4}\right)$

$\int \mathrm{xlog}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{{\mathrm{x}}^2}{4}\left(2\log \left(\mathrm{x}\right) - 1\right) + \mathrm{c}$

• $\frac{{\mathrm{x}}^2}{2}\left(2\log \left(\mathrm{x}\right) - 1\right)$

• $\frac{{\mathrm{x}}^2}{4}\left(2\log \left(\mathrm{x}\right) + 1\right) + \mathrm{c}$

• $\frac{{\mathrm{x}}^2}{2}\left(2\log \left(\mathrm{x}\right) + 1\right)$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}{1 - \sin \left(\mathrm{x}\right) . \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

If ${\int }_0^1{\tan }^{-1}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{p}$, then the value of ${\int }_0^1{\tan }^{-1}\left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)$dx is

• $\frac{\mathrm{\pi }}{4} + \mathrm{p}$

• $\frac{\mathrm{\pi }}{4} - \mathrm{p}$

• 1 + p

• 1 - p

368.

If f(x) = x , g(x) = sin(x), then $\int \mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to 

• sin(x) + c

• - cos(x) + c

• $\frac{{\mathrm{x}}^2}{2} + \mathrm{c}$

• x sin(x) + c

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\log \left(\csc \left(\mathrm{x}\right)\right)d\mathrm{x}$ is

• $\frac{\mathrm{\pi }}{2}\log \left(2\right)$

• $\mathrm{\pi log}\left(2\right)$

• $- \frac{\mathrm{\pi }}{2}\log \left(2\right)$

• $2\mathrm{\pi log}\left(2\right)$

$\int {\mathrm{e}}^{\tan \left(\mathrm{x}\right)}\left({\sec }^2\left(\mathrm{x}\right) + {\sec }^3\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

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

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

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

• $\left(1 + \tan \left(\mathrm{x}\right)\right){\mathrm{e}}^{\tan \left(\mathrm{x}\right)} + \mathrm{c}$