The value of ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left({\mathrm{x}}^3 + \mathrm{xcos}\left(\mathrm{x}\right) + {\tan }^5\left(\mathrm{x}\right) + 1\right)d\mathrm{x}$ is equal to

• 0

• 2

• $\mathrm{\pi }$

• None of these

The value of the integral ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sin \left(- 4\mathrm{x}\right)d\mathrm{x}$ is

• $- \frac{8}{3}$

• $\frac{3}{2}$

• $\frac{8}{3}$

• None of these

$\int {\left(\frac{\mathrm{x} + 2}{\mathrm{x} + 4}\right)}^2{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{\mathrm{x}}{\mathrm{x} + 4}\right) +\hspace{0.17em}\mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{\mathrm{x} + 2}{\mathrm{x} + 4}\right) +\hspace{0.17em}\mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{\mathrm{x} - 2}{\mathrm{x} + 4}\right) +\hspace{0.17em}\mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{2{\mathrm{xe}}^{\mathrm{x}}}{\mathrm{x} + 4}\right) +\hspace{0.17em}\mathrm{C}$

The value of ${\int }_3^5\frac{{\mathrm{x}}^2}{{\mathrm{x}}^2 - 4}d\mathrm{x}$

• $2 - {\log }_{\mathrm{e}}\left(\frac{15}{7}\right)$

• $2 + {\log }_{\mathrm{e}}\left(\frac{15}{7}\right)$

• 2 + 4${\log }_{\mathrm{e}}\left(3\right) - 4{\log }_{\mathrm{e}}\left(7\right) + 4{\log }_{\mathrm{e}}\left(5\right)$

• $2 - {\tan }^{-1}\left(\frac{5}{7}\right)$

${\int }_0^{\infty }\frac{d\mathrm{x}}{{\left(\mathrm{x} +\hspace{0.17em}\sqrt{{\mathrm{x}}^2 +\hspace{0.17em}1}\right)}^3}$ is equal to

• $\frac{3}{8}$

• $\frac{1}{8}$

• $\frac{- 3}{8}$

• None of these

If for every integer n, ${\int }_{\mathrm{n}}^{\mathrm{n} + 1}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = {\mathrm{n}}^2$, then the value of ${\int }_{- 2}^4\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• 16

• 14

• 19

• None of these

The value of integral ${\int }_0^{\mathrm{\pi }}\mathrm{xf}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$ is

• 0

• $\mathrm{\pi }{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{f}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$

• $\frac{\mathrm{\pi }}{4}{\int }_0^{\mathrm{\pi }}\mathrm{f}\left(\sin \left(\mathrm{x}\right)\right)d\mathrm{x}$

• None of these

$\underset{\mathrm{x}\to \infty }{\lim }\frac{{\int }_0^{2\mathrm{x}}{\mathrm{xe}}^{\mathrm{x}}\mathrm{dx}}{{\mathrm{e}}^{4{\mathrm{x}}^2}}$ equals

• 0

• $\infty$

• 2

• $\frac{1}{2}$

399.

The value of ${\int }_{- 2}^3\left|1 - {\mathrm{x}}^2\right|d\mathrm{x}$, is

• $\frac{1}{3}$

• 14/3

• 7/3

• 28/3

$\int \frac{1}{\tan \left(\mathrm{x}\right) + \cot \left(\mathrm{x}\right) + \sec \left(\mathrm{x}\right) + \csc \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{1}{2}\left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) + \mathrm{x} \right)$ + C

• $\frac{1}{2}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) - \mathrm{x} \right)$ + C

• $\frac{1}{2}\left(\cos \left(\mathrm{x}\right) - \mathrm{x} + \sin \left(\mathrm{x}\right) \right)$ + C

• None of the above