Mathematics : Integrals

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

• $\frac{8\mathrm{\pi }\sqrt{3}}{5}$

• $\frac{2\mathrm{\pi }\sqrt{3}}{9}$

• $\frac{4{\mathrm{\pi }}^2\sqrt{3}}{9}$

• $\frac{{\mathrm{\pi }}^2}{6\sqrt{3}}$

$\int \frac{5\mathrm{x} + 3}{{\mathrm{x}}^2\left({\mathrm{x}}^2 - 2\right)}\mathrm{dx} = ?$

• $\frac{3}{2\mathrm{x}} + \frac{\sqrt{13}}{2\sqrt{2}}\log \left(\left|\frac{\sqrt{2} - \mathrm{x}}{\sqrt{2} + \mathrm{x}}\right|\right) + \mathrm{C}$

• $\frac{3}{2\mathrm{x}} + \frac{13}{4\sqrt{2}}\log \left(\left|\frac{x + \sqrt{2}}{x - \sqrt{2}}\right|\right) + \mathrm{C}$

• $\frac{3}{2\mathrm{x}} + \frac{13}{4\sqrt{2}}\log \left(\left|\frac{\mathrm{x} - \sqrt{2}}{\mathrm{x} + \sqrt{2}}\right|\right) + \mathrm{C}$

• $\frac{3}{5}x + \frac{5}{3\sqrt{2}}\log \left(\left|\frac{x + \sqrt{2}}{x - \sqrt{2}}\right|\right) + \mathrm{C}$

$\mathrm{If} \mathrm{y} = {\tan }^{-1}\left(\left\{\frac{\mathrm{x}}{1 + \sqrt{1 - {\mathrm{x}}^2}}\right\}\right) + \sin \left(2{\tan }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $\frac{1 - 2\mathrm{x}}{2\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1 - 2\mathrm{x}}{\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{2\mathrm{x} + 1}{\mathrm{x}\sqrt{1 - \mathrm{x}}}$

• $\frac{2 - \mathrm{x}}{2\sqrt{1 - {\mathrm{x}}^2}}$

If ${\int }_0^{10}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}= 5, \mathrm{then} \sum _{\mathrm{k} = 1}^{10}{\int }_0^1\mathrm{f}\left(\mathrm{k} - 1 + \mathrm{x}\right)\mathrm{dx}$ = ?

• 50

• 10

• 5

• 20

$\int \frac{{\mathrm{x}}^{\mathrm{e} - 1} + {\mathrm{e}}^{\mathrm{x} - 1}}{{\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}$

• $\frac{- 1}{\mathrm{e}}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $- \mathrm{elog}\left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $\frac{1}{\mathrm{e}}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $\mathrm{e}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

$\int \frac{\mathrm{x}+ \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx} = ?$

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• $\mathrm{xsin}\left(\frac{\mathrm{x}}{2}\right) + \cos \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \sec \left(\frac{\mathrm{x}}{2}\right)\hspace{0.17em}+ \mathrm{C}$

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \sec \left(\frac{\mathrm{x}}{2}\right)\hspace{0.17em}+ \mathrm{C}$

$\mathrm{If} {\mathrm{I}}_{\mathrm{n}} = \int \frac{\sin \left(\mathrm{nx}\right)}{\cos \left(\mathrm{x}\right)}\mathrm{dx}, \mathrm{then} {\mathrm{I}}_{\mathrm{n}} = ?$

• $\frac{ - 2}{\mathrm{n} - 1}\cos \left(\mathrm{n} - 1\right)\mathrm{x} - {\mathrm{I}}_{\mathrm{n} - 2}$

• $\frac{2}{\mathrm{n} - 1}\cos \left(\mathrm{n} - 1\right)\mathrm{x} + {\mathrm{I}}_{\mathrm{n} - 2}$

• $\frac{ - 2}{\mathrm{n} + 1}\sin \left(\mathrm{n} - 1\right)\mathrm{x} - {\mathrm{I}}_{\mathrm{n} - 2}$

• $\frac{ - 2}{\mathrm{n} + 1}\cos \left(\mathrm{n} - 1\right)\mathrm{x} - {\mathrm{I}}_{\mathrm{n} - 2}$

$\mathrm{The} \mathrm{integral} {\int }_0^2\left|\left|\mathrm{x} - 1\right| - \mathrm{x}\right|d\mathrm{x} = ?$

Let [t] denote the greatest integer less than or equal to t. Then the value of ${\int }_1^2\left|2\mathrm{x} - \left[3\mathrm{x}\right]\right|d\mathrm{x}$ is

$\mathrm{Area} \mathrm{enclosed} \mathrm{by} 0 \le \mathrm{y} \le {\mathrm{x}}^2 + 1, 0 \le \mathrm{y} \le \mathrm{x} + 1, \frac{1}{2} \le \mathrm{x} \le 2 \mathrm{is} :$

• $\frac{1}{12}$

• $\frac{1}{6}$

• 1

• $\frac{1}{3}$