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

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

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

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

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

482.

$\int \frac{5\mathrm{x} \mathrm{dx}}{{\left(1 - \mathrm{x}\right)}^3}$ 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${\sin }^{-1}\left(\tan \left(\mathrm{x}\right)\right)$

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

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

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

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

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{C}$

• ${\mathrm{xe}}^{\mathrm{x}}\log \left|\mathrm{x}\right| + \mathrm{C}$

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

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

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

• $\frac{1}{1 + \mathrm{xlog}\left|\mathrm{x}\right|} + \mathrm{C}$

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

• $\frac{- 1}{1 + \mathrm{xlog}\left|\mathrm{x}\right|} + \mathrm{C}$

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

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

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

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

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

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

The value of ${\int }_0^6\left|\mathrm{x} - 3\right|d\mathrm{x}$ is equal to

• 6

• 0

• 12

• 9

If f(x) = ${\int }_{2\mathrm{x}}^{\sin \left(\mathrm{x}\right)}\cos \left({\mathrm{t}}^3\right)\mathrm{dt}$, then f'(x) is equal to

• $\cos \left({\sin }^3\left(\mathrm{x}\right)\right)\cos \left(\mathrm{x}\right) - 2\cos \left(8{\mathrm{x}}^3\right)$

• $\sin \left({\sin }^3\left(\mathrm{x}\right)\right)\sin \left(\mathrm{x}\right) - 2\sin \left(8{\mathrm{x}}^3\right)$

• $\cos \left({\cos }^3\left(\mathrm{x}\right)\right)\cos \left(\mathrm{x}\right) - 2\cos \left({\mathrm{x}}^3\right)$

• $\cos \left({\sin }^3\left(\mathrm{x}\right)\right) - \cos \left(8{\mathrm{x}}^3\right)$