$\mathrm{What} \mathrm{is} {\int }_0^1\mathrm{x}{\left(1 - \mathrm{x}\right)}^{9}d\mathrm{x} = ?$

• $\frac{1}{110}$

• $\frac{1}{132}$

• $\frac{1}{148}$

• $\frac{1}{240}$

$\mathrm{What} \mathrm{is} {\int }_2^8\left|\mathrm{x} - 5\right|d\mathrm{x} = ?$

• 2

• 3

• 4

• 9

What is $\int {\sin }^3\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)\mathrm{dx} = ?$

• ${\cos }^4\left(\mathrm{x}\right) + \mathrm{c}$

• ${\sin }^4\left(\mathrm{x}\right) + \mathrm{c}$

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

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

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

• $\ln \left|\tan \left(\mathrm{x}\right)\right| + \mathrm{c}$

• $\ln \left|\sec \left(\mathrm{x}\right)\right|$

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

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

$\mathrm{What} \mathrm{is} {\int }_{ - 1}^1\frac{\mathrm{d}}{\mathrm{dx}}\left({\tan }^{-1}\left(\frac{1}{\mathrm{x}}\right)\right)d\mathrm{x} = ?$

• 0

• $- \frac{\mathrm{\pi }}{4}$

• $- \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2}$

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

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

• $\mathrm{c} - \frac{1}{\mathrm{ab}}{\tan }^{-1}\left(\frac{\mathrm{b}\tan \left(\mathrm{x}\right)}{\mathrm{a}}\right)$

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

• None of the above

$\mathrm{What} \mathrm{is} \int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^7 = 1\right)} = ?$

• $\frac{1}{2}\ln \left(\left|\frac{{\mathrm{x}}^7 - 1}{{\mathrm{x}}^7 +\hspace{0.17em}1}\right|\right) + \mathrm{c}$

• $\frac{1}{7}\ln \left(\frac{{\mathrm{x}}^2 +\hspace{0.17em}1}{{\mathrm{x}}^7}\right) + \mathrm{c}$

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

• $\frac{1}{7}\ln \left(\left|\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right|\right) + \mathrm{c}$

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

• $\frac{{\mathrm{x}}^2}{2} + \mathrm{c}$

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

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

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

19.

$\mathrm{If} \mathrm{xdy} = \mathrm{y}(\mathrm{dx} + \mathrm{ydy}); \mathrm{y}\left(1\right) = 1 \mathrm{and} \mathrm{y}\left(\mathrm{x}\right) > 0, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{y}( - 3) \mathrm{equal} \mathrm{to} ?$

• 3

• 2

• 1

• 0

If f(x) and g(x) are continuous functions satisfying f(x) = f(a - x) and g(a) + g(a - x) = 2, then what is

${\int }_0^{\mathrm{a}}$f(x)g(x)dx equal to ?

• ${\int }_0^{\mathrm{a}}\mathrm{g}\left(\mathrm{x}\right)d\mathrm{x}$

• ${\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $2{\int }_{ 0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• 0