Mathematics : Integrals

The value of $\int \frac{\mathrm{dx}}{\sqrt{2\mathrm{x} - {\mathrm{x}}^2}}$ is

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

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

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

• $- \sqrt{2\mathrm{x} - {\mathrm{x}}^2} + \mathrm{c}$

The value of $\int \mathrm{xlog}\left({\mathrm{x}}^3\right)\mathrm{dx}$ is

• $\frac{1}{16}\left(4{\mathrm{x}}^4\log \left(\mathrm{x}\right) - {\mathrm{x}}^4 + \mathrm{c}\right)$

• $\frac{1}{8}\left({\mathrm{x}}^4\log \left(\mathrm{x}\right) - 4{\mathrm{x}}^4 + \mathrm{c}\right)$

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

• $\frac{1}{16}\left(4{\mathrm{x}}^4\log \left(\mathrm{x}\right) + {\mathrm{x}}^4 + \mathrm{c}\right)$

The value of ${\int }_0^{\mathrm{\pi }}\frac{\mathrm{dx}}{5 + 3\cos \left(\mathrm{x}\right)}$ is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{2}$

• zero

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\log \left(\tan \left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• - 1

• $\frac{1}{2}$

• zero

• 1

The value of ${\int }_{- \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\cos \left(\mathrm{x}\right)\left[\log \left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)\right]\mathrm{dx}$ is

• 2e^1/2

• 1

• e^1/2

• zero

The function f(x) = ${\int }_0^{\mathrm{x}}\log \left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right)\mathrm{dx}$ is

• an even function

• an odd function

• a periodic function

• None of these

$\int \frac{\sin \left(2\mathrm{x}\right)}{1 + {\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ =

• $- \frac{1}{2}\log \left(1 + {\cos }^2\left(\mathrm{x}\right)\right) + \mathrm{c}$

• $2\log \left(1 + {\cos }^2\left(\mathrm{x}\right)\right) + \mathrm{c}$

• $\frac{1}{2}\log \left(1 + \cos \left(2\mathrm{x}\right)\right) + \mathrm{c}$

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

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

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{1 + \sin \left(\mathrm{x}\right)}{1 - \cos \left(\mathrm{x}\right)}\right) + \mathrm{c}$

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

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

• ${\mathrm{e}}^{- \mathrm{x}}\tan \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{e}}^{- \mathrm{x}}\sec \left(\mathrm{x}\right) + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\sec \left(\mathrm{x}\right) + \mathrm{c}$

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

${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\csc }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• - 1

• 1

• 0

• $\frac{1}{2}$