The value of integral ${\int }_0^1\sqrt{\frac{1 - \mathrm{x}}{1 +\hspace{0.17em}\mathrm{x}}}d\mathrm{x}$ is

• $\frac{\mathrm{\pi }}{2} + 1$

• $\frac{\mathrm{\pi }}{2} - 1$

• - 1

• 1

Evaluate $\int \frac{{\mathrm{x}}^2 + 4}{{\mathrm{x}}^4 + 16}\mathrm{dx}$.

• $\frac{1}{2\sqrt{2}}{\tan }^{-1}\left(\frac{{\mathrm{x}}^2 - 4}{2\mathrm{x}\sqrt{2}}\right) + \mathrm{C}$

• $\frac{1}{2\sqrt{2}}{\tan }^{-1}\left(\frac{{\mathrm{x}}^2 - 4}{2\sqrt{2}}\right) + \mathrm{C}$

• $\frac{1}{2\sqrt{2}}{\tan }^{-1}\left(\frac{{\mathrm{x}}^2 - 4}{\mathrm{x}\sqrt{2}}\right) + \mathrm{C}$

• None of these

Evaluate ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{1}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$

• 2

• - 2

• 1/2

• - 1/2

If $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right), \mathrm{then} \int {\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^2\mathrm{dx}$ is equal to

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

• ${\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^3$

• $\frac{{\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^3}{3}$

• ${\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^2$

$\int {\sin }^{-1}\left\{\frac{\left(2\mathrm{x} +\hspace{0.17em}2\right)}{\sqrt{4{\mathrm{x}}^2 + 8\mathrm{x} + 13}}\right\}\mathrm{dx}$ is equal to

• $\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\log \left(\frac{4{\mathrm{x}}^2 + 8\mathrm{x} + 13}{9}\right) + \mathrm{c}$

• $\frac{3}{2}{\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\mathrm{og}\left(\frac{{\displaystyle 4{\mathrm{x}}^2 + 8\mathrm{x} + 13}}{{\displaystyle 9}}\right) + \mathrm{c}$

• $\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{2}\mathrm{l}\mathrm{og}\left(4{\mathrm{x}}^2 + 8\mathrm{x} + 13\right) + \mathrm{c}$

• $\frac{3}{2}\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\log \left(4{\mathrm{x}}^2 + 8\mathrm{x} + 13\right) + \mathrm{c}$

If ${\int }_1^{\mathrm{x}}\frac{d\mathrm{t}}{\left|\mathrm{t}\right|\sqrt{{\mathrm{t}}^2 - 1}} = \frac{\mathrm{\pi }}{6}$, then x can be equal to

• $\frac{2}{\sqrt{3}}$

• $\sqrt{3}$

• 2

• None of these

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

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

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

• $2{\mathrm{e}}^{\mathrm{x}}\cot \left(\mathrm{x}\right) + \mathrm{c}$

• $- 2{\mathrm{e}}^{\mathrm{x}}\cot \left(\mathrm{x}\right) + \mathrm{c}$

$\mathrm{If} {\mathrm{I}}_{\mathrm{n}} = \int {\sin }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}, \mathrm{then} {\mathrm{nI}}_{\mathrm{n}} - \left(\mathrm{n} - 1\right){\mathrm{I}}_{\mathrm{n} - 2}$ equals

• ${\sin }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\mathrm{xcos}\left(\mathrm{x}\right)$

• ${\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$

• $- {\sin }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)$

• $- {\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$

If I = $\int \frac{{\mathrm{x}}^5}{\sqrt{1 + {\mathrm{x}}^3}}\mathrm{dx}$, then I is equal to

• $\frac{2}{9}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{5}{2}} + \frac{2}{3}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{3}{2}} + \hspace{0.17em}\mathrm{c}$

• $\log \left|\sqrt{\mathrm{x}} + \sqrt{1 + {\mathrm{x}}^3}\right| + \mathrm{c}$

• $\log \left|\sqrt{\mathrm{x}} - \sqrt{1 + {\mathrm{x}}^3}\right| + \mathrm{c}$

• $\frac{2}{9}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{3}{2}} - \frac{2}{3}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{1}{2}} + \hspace{0.17em}\mathrm{c}$

140.

The value of ${\int }_0^{\mathrm{a}}\sqrt{\frac{\mathrm{a} - \mathrm{x}}{\mathrm{x}}}d\mathrm{x}$ is

• $\frac{\mathrm{a}}{2}$

• $\frac{\mathrm{a}}{4}$

• $\frac{\mathrm{\pi a}}{2}$

• $\frac{\mathrm{\pi a}}{4}$