Mathematics : Integrals

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

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

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

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

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

If f(x) = x - [x], for every real number x, where [x] is the integral part of x. Then ${\int }_{- 1}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• 1

• 2

• 0

• $\frac{1}{2}$

The value of integral

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

• $\log \left(\frac{4}{3}\right)$

• $4\log \left(\frac{3}{4}\right)$

• $4\log \left(\frac{4}{3}\right)$

• $\log \left(\frac{3}{4}\right)$

$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + \sqrt{2}}$ equals to

• $- \frac{1}{\sqrt{2}}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $\frac{1}{2}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $\frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $- \frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

The value of I = ${\int }_0^1\mathrm{x}\left|\mathrm{x} - \frac{1}{2}\right|\mathrm{dx}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• None of these

The value of integral ${\int }_0^1\sqrt{\frac{1 - \mathrm{x}}{1 +\hspace{0.17em}\mathrm{x}}}d\mathrm{x}$ is

• $\frac{\mathrm{\pi }}{2} + 1$

• $\frac{\mathrm{\pi }}{2} - 1$

• - 1

• 1

Evaluate $\int \frac{{\mathrm{x}}^2 + 4}{{\mathrm{x}}^4 + 16}\mathrm{dx}$.

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

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

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

• None of these

Evaluate ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{1}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$

• 2

• - 2

• 1/2

• - 1/2

If $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right), \mathrm{then} \int {\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^2\mathrm{dx}$ is equal to

• $\frac{1}{2}{\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^2$

• ${\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^3$

• $\frac{{\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^3}{3}$

• ${\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^2$

$\int {\sin }^{-1}\left\{\frac{\left(2\mathrm{x} +\hspace{0.17em}2\right)}{\sqrt{4{\mathrm{x}}^2 + 8\mathrm{x} + 13}}\right\}\mathrm{dx}$ is equal to

• $\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\log \left(\frac{4{\mathrm{x}}^2 + 8\mathrm{x} + 13}{9}\right) + \mathrm{c}$

• $\frac{3}{2}{\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\mathrm{og}\left(\frac{{\displaystyle 4{\mathrm{x}}^2 + 8\mathrm{x} + 13}}{{\displaystyle 9}}\right) + \mathrm{c}$

• $\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{2}\mathrm{l}\mathrm{og}\left(4{\mathrm{x}}^2 + 8\mathrm{x} + 13\right) + \mathrm{c}$

• $\frac{3}{2}\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\log \left(4{\mathrm{x}}^2 + 8\mathrm{x} + 13\right) + \mathrm{c}$