Mathematics : Integrals

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{{\tan }^7\left(\mathrm{x}\right)}{{\cot }^7\left(\mathrm{x}\right) + {\tan }^7\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

$\int {\mathrm{xsec}}^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\mathrm{xtan}\left(\mathrm{x}\right) + \log \left(\sec \left(\mathrm{x}\right)\right) + \mathrm{c}$

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

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

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

$\int \frac{\mathrm{t}}{{\mathrm{e}}^{3{\mathrm{t}}^2}}\mathrm{dt}$ is equal to

• $\frac{1}{6}{\mathrm{e}}^{3{\mathrm{t}}^2} + \mathrm{c}$

• $- \frac{1}{6}{\mathrm{e}}^{3{\mathrm{t}}^2} + \mathrm{c}$

• $\frac{1}{6}{\mathrm{e}}^{- 3{\mathrm{t}}^2} + \mathrm{c}$

• $- \frac{1}{6}{\mathrm{e}}^{- 3{\mathrm{t}}^2} + \mathrm{c}$

${\int }_0^{\mathrm{\pi }}\log \left({\sin }^2\left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• $2{\mathrm{\pi log}}_{\mathrm{e}}\left(\frac{1}{2}\right)$

• ${\mathrm{\pi log}}_{\mathrm{e}}\left(2\right)$

• $\frac{\mathrm{\pi }}{2}{\log }_{\mathrm{e}}\left(\frac{1}{2}\right)$

• None of these

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

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

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

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

• None of these

${\int }_0^2\left[{\mathrm{x}}^2\right]\mathrm{dx}$ is equal to

• $2 - \sqrt{2}$

• $2 + \sqrt{2}$

• $\sqrt{2} - 1$

• $- \sqrt{2} - \sqrt{3} + 5$

In = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$, then $\underset{\mathrm{n}\to \infty }{\lim }\mathrm{n}\left[{\mathrm{I}}_{\mathrm{n}} + {\mathrm{I}}_{\mathrm{n} + 2}\right]$ equals

• 1/ 2

• 2 sq units

• 3 sq units

• 4 sq uits

${\int }_0^{10\mathrm{\pi }}\left|\sin \left(\mathrm{x}\right)\right|\mathrm{dx}$ is equal to

• 20

• 8

• 10

• 18

If I = ${\int }_{{\mathrm{x}}_0}^{{\mathrm{x}}_0 + \mathrm{nh}}\mathrm{ydx}$, then by Trapezoidal rule I is equal to

• $\mathrm{h}\left[\left({\mathrm{y}}_0 + {\mathrm{y}}_{\mathrm{n}}\right) + 2\left({\mathrm{y}}_1 + {\mathrm{y}}_2 + ... + {\mathrm{y}}_{\mathrm{n} - 1}\right)\right]$

• $\mathrm{h}\left[\frac{1}{2}\left({\mathrm{y}}_0 + {\mathrm{y}}_{\mathrm{n}}\right) + 2\left({\mathrm{y}}_1 + {\mathrm{y}}_2 + ... + {\mathrm{y}}_{\mathrm{n} - 1}\right)\right]$

• $\frac{\mathrm{h}}{2}\left[\left({\mathrm{y}}_0 + {\mathrm{y}}_{\mathrm{n}}\right) + 2\left({\mathrm{y}}_1 + {\mathrm{y}}_2 + ... + {\mathrm{y}}_{\mathrm{n} - 1}\right)\right]$

• $\mathrm{h}\left[\left({\mathrm{y}}_0 + {\mathrm{y}}_{\mathrm{n}}\right) + 2\left({\mathrm{y}}_1 + {\mathrm{y}}_2 + ... + {\mathrm{y}}_{\mathrm{n} - 1}\right)\right]$

$\int \frac{\mathrm{dx}}{1 - {\mathrm{x}}^2}$ is equal to

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

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

• $\frac{1}{2}\log \left|\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right| + \mathrm{c}$