$\int \frac{1}{16{\mathrm{x}}^2 + 9}\mathrm{dx}$ is equal to

• $\frac{1}{3}{\tan }^{-1}\left(\frac{4\mathrm{x}}{3}\right) + \mathrm{c}$

• $\frac{1}{4}{\tan }^{-1}\left(\frac{4\mathrm{x}}{3}\right) + \mathrm{c}$

• $\frac{1}{12}{\tan }^{-1}\left(\frac{4\mathrm{x}}{3}\right) + \mathrm{c}$

• $\frac{1}{12}{\tan }^{-1}\left(\frac{3\mathrm{x}}{4}\right) + \mathrm{c}$

The value of ${\int }_4^7\frac{{\left(11 - \mathrm{x}\right)}^2}{{\mathrm{x}}^2 + {\left(11 - \mathrm{x}\right)}^2}\mathrm{dx}$ is

• 1

• 1/2

• 3/2

• 0

$\int \left(\sqrt{\tan \left(\mathrm{x}\right)} + \sqrt{\cot \left(\mathrm{x}\right)}\right)\mathrm{dx}$

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

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

• $\frac{\tan \left(\mathrm{x}\right)}{\sqrt{2}} . {\tan }^{-1}\left(\frac{\cot \left(\mathrm{x}\right) + 1}{\sqrt{2\tan \left(\mathrm{x}\right)}}\right) + \mathrm{C}$

• $\frac{\tan \left(\mathrm{x}\right)}{\sqrt{2}} . {\tan }^{-1}\left(\frac{{\displaystyle \cot }\left(\mathrm{x}\right){\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle 1}}{\sqrt{{\displaystyle 2}\tan \left(\mathrm{x}\right)}}\right) + \mathrm{C}$

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

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

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

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

• None of these

375.

${\int }_0^{\mathrm{x}}\frac{\mathrm{xdx}}{1 + \cos \left(\mathrm{\alpha }\right)\sin \left(\mathrm{x}\right)}, \left(0 < \mathrm{\alpha } < \mathrm{\pi }\right)$ is equal to

• $\frac{\mathrm{\pi \alpha }}{\sin \left(\mathrm{\alpha }\right)}$

• $\frac{\mathrm{\pi \alpha }}{\cos \left(\mathrm{\alpha }\right)}$

• $\frac{\mathrm{\pi \alpha }}{1 +\hspace{0.17em}\sin \left(\mathrm{\alpha }\right)}$

• $\frac{\mathrm{\pi \alpha }}{1 - \cos \left(\mathrm{\alpha }\right)}$

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\cos \left(\mathrm{x}\right)}{1 + {\mathrm{e}}^{\mathrm{x}}}d\mathrm{x}$

• 1

• 0

• - 1

• None of these

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

• $\mathrm{\pi }$

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

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

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

By Simpson rule taking n = 4, the value of the integral ${\int }_0^1\frac{1}{1 + {\mathrm{x}}^2}\mathrm{dx}$ is equal to

• 0.788

• 0.781

• 0.785

• None of the above

The value of ${\int }_0^{\mathrm{\pi }}\log \left(1 +\hspace{0.17em}\cos \left(\mathrm{x}\right)\right)\mathrm{dx}$ is

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

• $\mathrm{\pi log}\left(\frac{1}{2}\right)$

• $\mathrm{\pi log}\left(2\right)$

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

The value of ${\int }_3^4\sqrt{\left(4 - \mathrm{x}\right)\left(\mathrm{x} - 3\right)}\mathrm{dx}$ is

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$