Mathematics : Integrals

${\int }_2^3\frac{\mathrm{dx}}{{\mathrm{x}}^2 - \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to}}$

• $\log \left(\frac{2}{3}\right)$

• $\log \left(\frac{4}{3}\right)$

• $\log \left(\frac{8}{3}\right)$

• $\log \left(\frac{1}{4}\right)$

${\int }_{\raisebox{1ex}{$- \mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^4\left(\mathrm{x}\right){\cos }^6\left(\mathrm{x}\right)\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{3\mathrm{\pi }}{128}$

• $\frac{3\mathrm{\pi }}{256}$

• $\frac{3\mathrm{\pi }}{572}$

• $\frac{3\mathrm{\pi }}{64}$

The approximate value of ${\int }_2^9 {\mathrm{x}}^2\mathrm{dx}$ by using trapezoidal rule with 4 equal intervals, is

• 248

• 242.5

• 242.8

• 243

$\mathrm{A} \mathrm{minimum} \mathrm{value} \mathrm{of} {\int }_0^{\mathrm{x}}{\mathrm{te}}^{{\mathrm{t}}^2}\mathrm{dt} \mathrm{is}$

• 0

• 1

• 2

• 3

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

• $\frac{1}{2}\sqrt{1 + \mathrm{x}} + \mathrm{C}$

• $\frac{2}{3}{\left(1 + \mathrm{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + \mathrm{C}$

• $\sqrt{1 + \mathrm{x}} + \mathrm{C}$

• $2{\left(1\hspace{0.17em}+ \mathrm{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} +\mathrm{C}$

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

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

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

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

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

${\int }_{- 2}^2\left|\begin{array}{c}\left[\mathrm{x}\right]\end{array}\right|\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 1

• 2

• 3

• 4

${\int }_0^1\sin \left(2{\tan }^{-1}\left(\sqrt{\frac{1 + \mathrm{x}}{1 - \mathrm{x}}}\right)\right)\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

${\int }_0^3\frac{3\mathrm{x} + 1}{{\mathrm{x}}^2 + 9}\mathrm{dx}$ is equal to :

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{12}$

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{2}$

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{6}$

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{3}$

If [2, 6] is divided into four intervals of equal length, then the approximate value of ${\int }_2^6\frac{1}{{\mathrm{x}}^2 - \mathrm{x}}\mathrm{dx}$ using Simpson's rule, is

• 0.3222

• 0.2333

• 0.5222

• 0.2555