Mathematics : Integrals

The value of the integral ${\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\sqrt{\mathrm{x}}\mathrm{dx}}{\sqrt{\mathrm{x}} + \sqrt{\mathrm{a} + \mathrm{b} - \mathrm{x}}}$ is :

• $\mathrm{\pi }$

• $\frac{1}{2}\left(\mathrm{b} - \mathrm{a}\right)$

• $\mathrm{\pi }/2$

• b - a

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\sqrt{\cot \left(\mathrm{x}\right)}}{\sqrt{\cot \left(\mathrm{x}\right)} + \sqrt{\tan \left(\mathrm{x}\right)}}\mathrm{dx}$ is equal to :

• 1

• - 1

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathrm{xsin}}^2\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to :

• $\frac{{\mathrm{\pi }}^2}{32}$

• $\frac{{\mathrm{\pi }}^2}{16}$

• $\frac{\mathrm{\pi }}{32}$

• None of these

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{\mathrm{xsin}\left(\mathrm{x}\right)}{{\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is :

• $\frac{1}{3}\left(4\mathrm{\pi } + 1\right)$

• $\frac{4\mathrm{\pi }}{3} - 2\log \left(\tan \left(\frac{5\mathrm{\pi }}{12}\right)\right)$

• $\frac{4\mathrm{\pi }}{3} + \log \left(\tan \left(\frac{5\mathrm{\pi }}{12}\right)\right)$

• None of these

$\int \frac{{\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^3}{\left(1 + {\mathrm{x}}^2\right)}\mathrm{dx}$ is equa to :

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

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

• ${\left({\tan }^{-1}\left(\mathrm{x}\right)\right)}^4 + \mathrm{c}$

• None of these

${\int }_0^{\mathrm{\pi }}\frac{\mathrm{xdx}}{1 + \sin \left(\mathrm{x}\right)}$ is equal to :

• $- \mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• None of these

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

• $\frac{1}{5}\log \left({\mathrm{x}}^5\right)\left({\mathrm{x}}^5 + 1\right) + \mathrm{c}$

• $\frac{1}{5}\log \left(\frac{{\mathrm{x}}^5 + 1}{{\mathrm{x}}^5}\right) + \mathrm{c}$

• $\frac{1}{5}\log \left(\frac{{\mathrm{x}}^5}{{\mathrm{x}}^5 + 1}\right) + \mathrm{c}$

• None of these

$\int \frac{\mathrm{x} + \sin \left(\mathrm{x}\right)}{1 +\hspace{0.17em}\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• ${\mathrm{xsec}}^2\left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

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

• None of these

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}{\cos }^3\left(4\mathrm{\theta }\right)\mathrm{d\theta }$ is equal to

• $\frac{5}{3}$

• $\frac{5}{4}$

• $\frac{1}{3}$

• $\frac{1}{6}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right) \sin \left(\mathrm{x}\right)\hspace{0.17em}}\mathrm{dx}$ is equal to

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$