${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{\mathrm{dx}}{1 + \sqrt{\tan \left(\mathrm{x}\right)}}$ is equal to

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

If f is a continuous function, then

• ${\int }_{- 2}^2\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_0^2\left[\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}(- \mathrm{x})\right]\mathrm{dx}$

• ${\int }_{- 3}^{5}2\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 6}^{10}\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

• ${\int }_{- 3}^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 4}^4\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

• ${\int }_{- 3}^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 2}^6\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

An integrating factor of the differential equation

$\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}\mathrm{ylog}\left(\mathrm{x}\right) = {\mathrm{xe}}^{\mathrm{x}}{\mathrm{x}}^{\frac{1}{2}\log \left(\mathrm{x}\right)}$, (x > 0), is

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

• ${\left(\sqrt{\mathrm{x}}\right)}^{\log \left(\mathrm{x}\right)}$

• ${\left(\sqrt{\mathrm{e}}\right)}^{{\left(\log \left(\mathrm{x}\right)\right)}^2}$

• ${\mathrm{e}}^{{\mathrm{x}}^2}$

Solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}}\tan \left(\mathrm{y}\right) = \sin \left(\mathrm{x} + \mathrm{y}\right) + \sin \left(\mathrm{x} - \mathrm{y}\right)$

• $\sec \left(\mathrm{y}\right) + 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

• $\sec \left(\mathrm{y}\right) - 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

• $\cos \left(\mathrm{y}\right) - 2\sin \left(\mathrm{x}\right) = \mathrm{c}$

• $\tan \left(\mathrm{y}\right) - 2\sec \left(\mathrm{y}\right) = \mathrm{c}$

25.

Four persons are chosen at random from a group of 3 men, 2 women and 4 children. The chance that exactly 2 of them are children, is 

• $\frac{10}{21}$

• $\frac{11}{13}$

• $\frac{13}{25}$

• $\frac{21}{32}$