What is $\int {\mathrm{e}}^{\mathrm{xln}\left(\mathrm{a}\right)}\mathrm{dx} = ?$

• $\frac{{\mathrm{a}}^{\mathrm{x}}}{\ln \left(\mathrm{a}\right)} +\hspace{0.17em}\mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\ln \left(\mathrm{a}\right)} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\ln \left(\mathrm{ae}\right)} + \mathrm{c}$

• $\frac{{\mathrm{ae}}^{\mathrm{x}}}{\ln \left(\mathrm{a}\right)} + \mathrm{c}$

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left|\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right|d\mathrm{x} = ?$

• 0

• $2\left(\sqrt{2} - 1\right)$

• $2\sqrt{2}$

• $2\left(\sqrt{2} + 1\right)$

${\int }_0^{\frac{\mathrm{\pi }}{2}}{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}\cos \left(\mathrm{x}\right)d\mathrm{x} = ?$

• e  + 1

• e - 1

• e + 2

• e

$\mathrm{What} \mathrm{is} \int \ln \left({\mathrm{x}}^2\right)\mathrm{dx} = ?$

• $2\mathrm{xln}\left(\mathrm{x}\right) - 2\mathrm{x} + \mathrm{c}$

• $\frac{2}{\mathrm{x}} + \mathrm{c}$

• $2\mathrm{xln}\left(\mathrm{x}\right) + \mathrm{c}$

• $\frac{2\ln \left(\mathrm{x}\right)}{\mathrm{x}} - 2\mathrm{x} +\hspace{0.17em}\mathrm{c}$

$\mathrm{What} \mathrm{is} {\int }_0^{\mathrm{\pi }}{\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right)d\mathrm{x} = ?$

• $\frac{{\mathrm{e}}^{\mathrm{\pi }} + 1}{2}$

• $\frac{{\mathrm{e}}^{\mathrm{\pi }} - 1}{2}$

• ${\mathrm{e}}^{\mathrm{\pi }} + 1$

• $\frac{{\mathrm{e}}^{\mathrm{\pi }} +\hspace{0.17em}1}{4}$

$\mathrm{What} \mathrm{is} {\int }_1^{\mathrm{e}}\mathrm{xln}\left(\mathrm{x}\right)d\mathrm{x} = ?$

• $\frac{\mathrm{e} + 1}{4}$

• $\frac{{\mathrm{e}}^2 + 1}{2}$

• $\frac{\mathrm{e} - 1}{2}$

• $\frac{{\mathrm{e}}^2 - 1}{4}$

$\mathrm{What} \mathrm{is} {\int }_0^{\sqrt{2}}\left[{\mathrm{x}}^2\right]d\mathrm{x} \mathrm{equal} \mathrm{to} (\mathrm{where} [.] \mathrm{is} \mathrm{the} \mathrm{greatest} \mathrm{integer} \mathrm{function}) ?$

• $\sqrt{2} - 1$

• $1 - \sqrt{2}$

• $2\left(\sqrt{2} - 1\right)$

• $\sqrt{3} - 1$

$\mathrm{What} \mathrm{is} \int \frac{\mathrm{dx}}{2^{\mathrm{x}} - 1} = ?$

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

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

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

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

$\mathrm{What} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} {\int }_{\raisebox{1ex}{$ - \mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\left(\sin \left(\mathrm{x}\right) - \tan \left(\mathrm{x}\right)\right)d\mathrm{x} ?$

• $- \frac{1}{\sqrt{2}} + \ln \left(\frac{1}{\sqrt{2}}\right)$

• $\frac{1}{\sqrt{2}}$

• 0

• $\sqrt{2}$

$\mathrm{If} {\int }_{\mathrm{a}}^{\mathrm{b}}{\mathrm{x}}^{3}d\mathrm{x} = 0 \mathrm{and} {\int }_{\mathrm{a}}^{\mathrm{b}}{\mathrm{x}}^{3}d\mathrm{x} = \frac{2}{3}, \mathrm{then} \mathrm{what} \mathrm{are} \mathrm{the} \mathrm{values} \mathrm{of} \mathrm{a} \mathrm{and} \mathrm{b} \mathrm{respectivley} ?$

•  - 1, 1

• 1, 1

• 0, 0

• 2, - 2