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

204.

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

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

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

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

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

${\int }_0^{10}\frac{{\mathrm{x}}^{10}}{{\left(10 - \mathrm{x}\right)}^{10} + {\mathrm{x}}^{10}}\mathrm{dx}$ is equal to

• 10

• 5

• 2

• $\frac{1}{2}$

The value of ${\int }_0^1\sqrt{\mathrm{x}}{\mathrm{e}}^{\sqrt{\mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{\left(\mathrm{e} - 2\right)}{2}$

• 2(e - 2)

• 2e - 1

• 2(e - 1)

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

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

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

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

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

$\int \frac{3^{\mathrm{x}}}{\sqrt{1 - 9^{\mathrm{x}}}}$dx is equal to

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

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

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

• $\frac{1}{9}{\sin }^{-1}\left(3^{\mathrm{x}}\right) +\hspace{0.17em}\mathrm{C}$

The value of the integral $\int \frac{\cos \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

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

• $\frac{1}{2}\left[\mathrm{x} + \log \left|\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right|\right] +\hspace{0.17em}\mathrm{C}$

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

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