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

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

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

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

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

192.

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

${\int }_0^1{\mathrm{xe}}^{- 5\mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{1}{25} - \frac{6{\mathrm{e}}^{- 5}}{25}$

• $\frac{1}{25} + \frac{6{\mathrm{e}}^{- 5}}{25}$

• $- \frac{1}{25} - \frac{6{\mathrm{e}}^{- 5}}{25}$

• $\frac{1}{25} + \frac{1}{25}{\mathrm{e}}^{- 5}$

$\int \frac{5\mathrm{x} \mathrm{dx}}{{\left(1 - \mathrm{x}\right)}^3}$ is equal to

• $\frac{5}{2{\left(\mathrm{x} - 1\right)}^2} - \frac{5}{\left(\mathrm{x} - 1\right)} + \mathrm{C}$

• $\frac{5}{2{\left(\mathrm{x} - 1\right)}^2} + \frac{5}{\left(\mathrm{x} - 1\right)} + \mathrm{C}$

• $\frac{5}{3{\left(\mathrm{x} - 1\right)}^2} + \frac{5}{2\left(\mathrm{x} - 1\right)} + \mathrm{C}$

• $\frac{5}{3{\left(\mathrm{x} - 1\right)}^2} - \frac{5}{2\left(\mathrm{x} - 1\right)} + \mathrm{C}$

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

• $2\log \left|\sqrt{\mathrm{x}} - 1\right| + \mathrm{C}$

• $2\log \left|\sqrt{\mathrm{x}} + 1\right| + \mathrm{C}$

• $\log \left|\sqrt{\mathrm{x}} - 1\right| + \mathrm{C}$

• $\frac{1}{2}\log \left|\sqrt{\mathrm{x}} + 1\right| + \mathrm{C}$

$\int \frac{\mathrm{dx}}{4{\sin }^2\left(\mathrm{x}\right) + 3{\cos }^2\left(\mathrm{x}\right)}$

• $\frac{\sqrt{3}}{4}{\tan }^{-1}\left(\frac{2\tan \left(\mathrm{x}\right)}{\sqrt{3}}\right) + \mathrm{C}$

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

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

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

$\int \frac{\sec \left(\mathrm{x}\right)\mathrm{dx}}{\sqrt{\cos \left(2\mathrm{x}\right)}}$ is equal to

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

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

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

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