${\int }_{- 1}^1\left({\mathrm{ax}}^3 + \mathrm{bx}\right)\mathrm{dx} = 0$ for

• any value of a and b

• a > 0, b > 0 only

• a > 0, b > 0 only

• a < 0, b > 0 only

Using the Trapezoidal rule, the approximate value of ${\int }_1^4\mathrm{ydx}$

x	1	2	3	4
y	0.7111	0.7222	0.7333	0.7444

• 0.1833

• 1.1833

• 2.1833

• 3.1833

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

• $\log \left(\left|\begin{array}{c}1 + \cot \left(\frac{\mathrm{x}}{2}\right)\end{array}\right|\right) + \mathrm{c}$

• $\log \left(\left|\begin{array}{c}1 - \tan \left(\frac{\mathrm{x}}{2}\right)\end{array}\right|\right) + \mathrm{c}$

• $\log \left(\left|\begin{array}{c}1 - \cot \left(\frac{\mathrm{x}}{2}\right)\end{array}\right|\right) + \mathrm{c}$

• $\log \left(\left|\begin{array}{c}1 + \tan \left(\frac{\mathrm{x}}{2}\right)\end{array}\right|\right) + \mathrm{c}$

$\int \frac{\mathrm{dx}}{7 + 5\cos \left(\mathrm{x}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{\sqrt{3}}{\tan }^{-1}\left(\frac{{\displaystyle 1}}{{\displaystyle \sqrt{3}}}\tan \frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $\frac{1}{\sqrt{6}}{\tan }^{-1}\left(\frac{{\displaystyle 1}}{{\displaystyle \sqrt{6}}}\tan \frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $\frac{1}{7}{\tan }^{-1}\left(\tan \frac{\mathrm{x}}{2}\right) + \mathrm{c}$

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

$\int \frac{3^{\mathrm{x}}\mathrm{dx}}{\sqrt{9^{\mathrm{x}} - 1}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{\log \left(3\right)}\log \left(\left|\begin{array}{c}{3}^{\mathrm{x}} + \sqrt{9^{\mathrm{x}} - 1}\end{array}\right|\right) + \mathrm{c}$

• $\frac{1}{\log \left(3\right)}\log \left(\left|\begin{array}{c}{3}^{\mathrm{x}} - \sqrt{9^{\mathrm{x}} - 1}\end{array}\right|\right) + \mathrm{c}$

• $\frac{1}{\log \left(9\right)}\log \left(\left|\begin{array}{c}{3}^{\mathrm{x}} + \sqrt{9^{\mathrm{x}} - 1}\end{array}\right|\right) + \mathrm{c}$

• $\frac{1}{\log \left(3\right)}\log \left(\left|\begin{array}{c}{9}^{\mathrm{x}} + \sqrt{9^{\mathrm{x}} - 1}\end{array}\right|\right) + \mathrm{c}$

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

• $\log \left(\frac{2}{3}\right)$

• $\log \left(\frac{4}{3}\right)$

• $\log \left(\frac{8}{3}\right)$

• $\log \left(\frac{1}{4}\right)$

${\int }_{\raisebox{1ex}{$- \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}$