Mathematics : Integrals

The value of $\int \frac{{\mathrm{x}}^2}{1 +\hspace{0.17em}{\mathrm{x}}^6}\mathrm{dx}$ is

• x³ + c

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

• log(1 + x³) + c

• None of these

$\int \left(1 - \cos \left(\mathrm{x}\right)\right){\csc }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

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

• $- \cot \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

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

• $- 2\cot \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

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

• $\sin \left(\frac{\mathrm{x}}{2}\right) - \cos \left(\frac{\mathrm{x}}{2}\right)$

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

• $2\left[\sin \left(\frac{\mathrm{x}}{2}\right) - \cos \left(\frac{\mathrm{x}}{2}\right)\right] +\hspace{0.17em}\mathrm{c}$

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

${\int }_1^3\frac{\cos \left(\log \left(\mathrm{x}\right)\right)}{\mathrm{x}}\mathrm{dx}$ is equal to

• 1

• $\cos \left(\log \left(3\right)\right)$

• $\sin \left(\log \left(3\right)\right)$

• $\frac{\mathrm{\pi }}{4}$

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

• $- \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

${\int }_1^2{\mathrm{e}}^{\mathrm{x}}\left(\frac{1}{\mathrm{x}} - \frac{1}{{\mathrm{x}}^2}\right)\mathrm{dx}$ is equal to

• $\mathrm{e} - \frac{{\mathrm{e}}^2}{2}$

• $\frac{{\mathrm{e}}^2}{2} - \mathrm{e}$

• $\frac{{\mathrm{e}}^2}{2} + \mathrm{e}$

• $\frac{{\mathrm{e}}^2}{2} - 2$

The value of ${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}{\sin }^3\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• 1

• 2

• 3

• 0

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

• $2\sqrt{{\mathrm{x}}^2 + 1} + {\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\sqrt{{\mathrm{x}}^2 - 1} - {\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $2\sqrt{{\mathrm{x}}^2 - 1} + {\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\frac{\sqrt{{\mathrm{x}}^2 - 1}}{2} + {\sin }^{-1}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

The value of ${\int }_{- 1}^1\log \left(\frac{\mathrm{x} - 1}{\mathrm{x} +\hspace{0.17em}1}\right)\mathrm{dx}$ is

• 1

• 2

• 0

• 4

Considering four sub-intervals, the value of ${\int }_0^4{2}^{\mathrm{x}}\mathrm{dx}$  by Simpson's rule is

• $\frac{64}{8}$

• $\frac{65}{3}$

• $\frac{62}{12}$

• $\frac{61}{8}$