31.

The distance between the X-axis and the point (3, 12, 5) is

• 3

• 13

• 14

• 12

If f(x) = ${\int }_1^{\mathrm{x}}{\sin }^2\left(\frac{\mathrm{t}}{2}\right)\mathrm{dt}$, then the value of $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{\pi } +\hspace{0.17em}\mathrm{x}\right) - \mathrm{f}\left(\mathrm{\pi }\right)}{\mathrm{x}}$ is equal to

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{3}{4}$

• 1

$\int \frac{1}{{\mathrm{x}}^2{\left({\mathrm{x}}^4 + 1\right)}^{{\displaystyle \frac{3}{4}}}}\mathrm{dx}$ is equal to

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

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

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{1}{4}}}}{\mathrm{x}} + \mathrm{C}$

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

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

• $- \cot \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{C}$

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

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

• $- \cot \left({\mathrm{xe}}^{\mathrm{x}}\right) + \mathrm{C}$

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

• $\frac{- {\mathrm{e}}^{\mathrm{x}}}{\mathrm{x} + 1} + \mathrm{C}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x} + 1} + \mathrm{C}$

• $\frac{{\mathrm{xe}}^{\mathrm{x}}}{\mathrm{x} + 1} + \mathrm{C}$

• $\frac{- {\mathrm{xe}}^{\mathrm{x}}}{\mathrm{x} + 1} + \mathrm{C}$

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

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

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

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

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

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

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

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

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

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

$\int \frac{\sqrt{{\mathrm{x}}^2 - 1}}{\mathrm{x}}\mathrm{dx}$ is equal to

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

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

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

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

$\int \frac{\sqrt{5 + {\mathrm{x}}^2}}{{\mathrm{x}}^4}\mathrm{dx}$ is equal to

• $\frac{1}{15}{\left(1 + \frac{5}{{\mathrm{x}}^2}\right)}^{\frac{3}{2}}$ + C

• $\frac{- 1}{15}{\left(1 + \frac{1}{{\mathrm{x}}^2}\right)}^{\frac{3}{2}} + \mathrm{C}$

• $\frac{- 1}{15}{\left(1 + \frac{5}{{\mathrm{x}}^2}\right)}^{\frac{3}{2}}$

• $\frac{1}{15}{\left(1 + \frac{1}{{\mathrm{x}}^2}\right)}^{\frac{3}{2}}$

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

• $\frac{1}{\mathrm{e}}\log \left(\frac{1 + \mathrm{e}}{2}\right)$

• $\log \left(\frac{1 + \mathrm{e}}{2}\right)$

• $\frac{1}{\mathrm{e}}\log \left(1 + \mathrm{e}\right)$

• $\log \left(\frac{2}{1 + \mathrm{e}}\right)$