Evaluate ${\int }_{- 2}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$, where f(x) = $$\begin{gathered} \left\{1 - 2\mathrm{x}, \mathrm{x} \le 0 \\ 1 + 2\mathrm{x}, \mathrm{x} \ge 0\right\} \end{gathered}$$

• 0

• 2

• 4

• 6

$\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x}}\left(\mathrm{x} +\hspace{0.17em}9\right)}$ is equal to

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

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

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

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

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

• xe^x + C

• x²x^x + C

• (x + 1)e^x + C

• (x² + 1)e^x + C

904.

$\int \frac{\mathrm{dx}}{{\mathrm{a}}^2{\sin }^2\left(\mathrm{x}\right) + {\mathrm{b}}^2{\cos }^2\left(\mathrm{x}\right)}$ is equal to

• $\frac{1}{\mathrm{ab}}{\tan }^{-1}\left(\frac{\mathrm{atan}\left(\mathrm{x}\right)}{\mathrm{b}}\right) + \mathrm{C}$

• ${\tan }^{-1}\left(\frac{\mathrm{atan}\left(\mathrm{x}\right)}{\mathrm{b}}\right) + \mathrm{C}$

• $\frac{1}{\mathrm{ab}}{\tan }^{-1}\left(\frac{\mathrm{btan}\left(\mathrm{x}\right)}{\mathrm{a}}\right) + \mathrm{C}$

• ${\tan }^{-1}\left(\frac{\mathrm{btan}\left(\mathrm{x}\right)}{\mathrm{a}}\right) + \mathrm{C}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^8\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{512}$

• $\frac{3\mathrm{\pi }}{512}$

• $\frac{5\mathrm{\pi }}{512}$

• $\frac{7\mathrm{\pi }}{512}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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