Mathematics : Integrals

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

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

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

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

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

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

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

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

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

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

If $\int \frac{\mathrm{f}\left(\mathrm{x}\right)}{\log \left(\cos \left(\mathrm{x}\right)\right)}\mathrm{dx} = - \log \left(\log \left(\cos \left(\mathrm{x}\right)\right)\right) + \mathrm{C}$, then f(x) is equal to

• $\tan \left(\mathrm{x}\right)$

• - $\sin \left(\mathrm{x}\right)$

• $- \cos \left(\mathrm{x}\right)$

• $- \tan \left(\mathrm{x}\right)$

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

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

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

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

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

$\int \frac{4{\mathrm{e}}^{\mathrm{x}} - 6{\mathrm{e}}^{- \mathrm{x}}}{9{\mathrm{e}}^{\mathrm{x}} - 4{\mathrm{e}}^{- \mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{3}{2}\mathrm{x} + \frac{35}{36}\log \left|9{\mathrm{e}}^{2\mathrm{x}} - 4\right| + \mathrm{C}$

• $\frac{3}{2}\mathrm{x} - \frac{35}{36}\log \left|9{\mathrm{e}}^{2\mathrm{x}} - 4\right| + \mathrm{C}$

• $- \frac{3}{2}\mathrm{x} + \frac{35}{36}\log \left|9{\mathrm{e}}^{2\mathrm{x}} - 4\right| + \mathrm{C}$

• $- \frac{5}{2}\mathrm{x} + \frac{35}{36}\log \left|9{\mathrm{e}}^{2\mathrm{x}} - 4\right| + \mathrm{C}$

$\int \sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\mathrm{dx}$ is equal to

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

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

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

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

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

• $\frac{1}{2} + \frac{1}{2}\log \left|\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $\frac{\mathrm{x}}{2} + \frac{1}{2}\log \left|\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $\frac{1}{2} + \frac{1}{2}\log \left|\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $\frac{\mathrm{x}}{2} + \frac{1}{2}\log \left|\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right)\right| + \mathrm{C}$

If ${\int }_0^{\mathrm{a}}\frac{{\mathrm{x}}^2 - 1}{1 - \mathrm{x}}\mathrm{dx} = - \frac{1}{2}$, then the value of a is equal to

• - 1

• 1

• 2

• - 2

The value of the integral ${\int }_0^1\mathrm{x}{\left(1 - \mathrm{x}\right)}^5\mathrm{dx}$ is equal to

• $\frac{1}{6}$

• $\frac{1}{7}$

• $\frac{6}{7}$

• $\frac{1}{42}$

If [x] denotes the greatest integer less than or equal to x, then the value of ${\int }_0^2\left(\left|\mathrm{x} - 2\right| + \left[\mathrm{x}\right]\right)\mathrm{dx}$ is equal to

• 2

• 3

• 1

• 4