The value of ${\int }_0^{\mathrm{\pi }}\left|\cos \left(\mathrm{x}\right)\right|d\mathrm{x}$ is

• $2\mathrm{\pi }$

• 2

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

• $\mathrm{\pi }$

The value of ${\int }_{- 3}^3\left({\mathrm{ax}}^5 + {\mathrm{bx}}^3 + \mathrm{cx} + \mathrm{k}\right)d\mathrm{x}$ where a, b, c, k are constant, depends only on

• a and k

• a and b

• a, b and c

• k

The value of the integral ${\int }_{- \mathrm{a}}^{\mathrm{a}}\frac{{\mathrm{xe}}^{{\mathrm{x}}^2}}{1 + {\mathrm{x}}^2}d\mathrm{x}$ is

• ${\mathrm{e}}^{{\mathrm{a}}^2}$

• 0

• ${\mathrm{e}}^{- {\mathrm{a}}^2}$

• a

Evaluate $\int \frac{{\mathrm{x}}^2}{\mathrm{x}\left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)}\mathrm{dx}$

If $\int \frac{\sin \left(\mathrm{x}\right)}{\sin \left(\mathrm{x} - \mathrm{a}\right)}\mathrm{dx} = \mathrm{Ax} + \mathrm{Blog}\left(\sin \left(\mathrm{x} - \mathrm{\alpha }\right)\right) + {\mathrm{c}}_1$, then the values of (A, B) is

• $\left(\sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)\right)$

• $\left(\cos \left(\mathrm{\alpha }\right), \sin \left(\mathrm{\alpha }\right)\right)$

• $\left(- \sin \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\alpha }\right)\right)$

• $\left(- \cos \left(\mathrm{\alpha }\right), \sin \left(\mathrm{\alpha }\right)\right)$

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

107.

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