21.

The value of x satisfying the equation ${\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\frac{2}{3}\right) + {\tan }^{-1}\left(\frac{7}{4}\right)$ is equal to

• $\frac{1}{2}$

• $- \frac{1}{2}$

• $\frac{3}{2}$

• $- \frac{1}{3}$

If ${\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) = \frac{2\mathrm{\pi }}{3}$, then ${\cot }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\mathrm{y}\right)$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{1}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\sqrt{3}}{2}$

$\int {\left(\sec \left(\mathrm{x}\right)\right)}^{\mathrm{m}}\left({\tan }^3\left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• ${\sec }^{\mathrm{m} + 2}\left(\mathrm{x}\right) + \mathrm{C}$

• ${\tan }^{\mathrm{m} +\hspace{0.17em}2}\left(\mathrm{x}\right) + \mathrm{C}$

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

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

$\int \frac{1}{7}\sin \left(\frac{\mathrm{x}}{7} + 10\right)\mathrm{dx}$ is equal to

• $\frac{1}{7}\cos \left(\frac{\mathrm{x}}{7} + 10\right) + \mathrm{C}$

• $- \frac{1}{7}\cos \left(\frac{\mathrm{x}}{7} + 10\right)\mathrm{dx}$

• $- \cos \left(\frac{\mathrm{x}}{7} + 10\right) + \mathrm{C}$

• $- 7\cos \left(\frac{\mathrm{x}}{7} + 10\right) + \mathrm{C}$

$\int \left(\frac{\mathrm{x} - \mathrm{a}}{\mathrm{x}} - \frac{\mathrm{x}}{\mathrm{x} +\hspace{0.17em}\mathrm{a}}\right)\mathrm{dx}$ is equal to

• $\log \left|\frac{\mathrm{x} + \mathrm{a}}{\mathrm{x}}\right| + \mathrm{C}$

• $\mathrm{alog}\left|\frac{\mathrm{x} + \mathrm{a}}{\mathrm{x}}\right| + \mathrm{C}$

• $\mathrm{alog}\left|\frac{\mathrm{x}}{\mathrm{x} +\hspace{0.17em}\mathrm{a}}\right| + \mathrm{C}$

• $\log \left|\frac{\mathrm{x} + \mathrm{a}}{\mathrm{x}}\right| + \mathrm{C}$

$\int {\mathrm{x}}^4{\mathrm{e}}^{{\mathrm{x}}^5}\cos \left({\mathrm{e}}^{{\mathrm{x}}^5}\right)\mathrm{dx}$ is equal to

• $\frac{1}{3}\sin \left({\mathrm{e}}^{{\mathrm{x}}^5}\right) + \mathrm{C}$

• $\frac{1}{4}\sin \left({\mathrm{e}}^{{\mathrm{x}}^5}\right) + \mathrm{C}$

• $\frac{1}{5}\sin \left({\mathrm{e}}^{{\mathrm{x}}^5}\right) + \mathrm{C}$

• $\sin \left({\mathrm{e}}^{{\mathrm{x}}^5}\right) + \mathrm{C}$

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

• $\log \left|\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $\log \left|\sin \left(2\mathrm{x}\right)\right| + \mathrm{C}$

• $\log \left|\sec \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $\log \left|\cos \left(\mathrm{x}\right)\right| + \mathrm{C}$

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

• $\mathrm{x} + \log \left(\tan \left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\mathrm{xlog}\left|\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

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

• $\mathrm{x} + \tan \left(\mathrm{x}\right) + \mathrm{C}$

$\int \frac{1}{8{\sin }^2\left(\mathrm{x}\right) + 1}\mathrm{dx}$ is equal to

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

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

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

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

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

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

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