${\int }_0^1{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ =

• $\frac{\mathrm{\pi }}{4} + \frac{1}{2}$

• $\frac{\mathrm{\pi }}{4} - \frac{1}{2}$

• $\frac{1}{2} - \frac{\mathrm{\pi }}{4}$

• $- \frac{\mathrm{\pi }}{4} - \frac{1}{2}$

If $\int \frac{1}{\sqrt{9 - 16{\mathrm{x}}^2}}\mathrm{dx} = {\mathrm{\alpha sin}}^{-1}\left(\mathrm{\beta x}\right) + \mathrm{c}$, then $\mathrm{\alpha } + \frac{1}{\mathrm{\beta }}$ =

• 1

• $\frac{7}{12}$

• $\frac{19}{12}$

• $\frac{9}{12}$

683.

If ${\int }_0^{\frac{\mathrm{\pi }}{2}}\log \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx} = \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$, then ${\int }_0^{\frac{\mathrm{\pi }}{2}}\log \left(\sec \left(\mathrm{x}\right)\right)\mathrm{dx}$ =

• $\frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

• $1 - \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

• $1 + \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

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

$\int \frac{1}{\left({\mathrm{x}}^2 + 4\right)\left({\mathrm{x}}^2 + 9\right)}\mathrm{dx} = {\mathrm{Atan}}^{-1}\left(\frac{\mathrm{x}}{2}\right) + {\mathrm{Btan}}^{-1}\left(\frac{\mathrm{x}}{3}\right) + \mathrm{C}$, then A - B =

• $\frac{1}{6}$

• $\frac{1}{30}$

• $- \frac{1}{30}$

• $- \frac{1}{6}$

If $\sqrt{\frac{\mathrm{x} - 5}{\mathrm{x} - 7}}\mathrm{dx} = \mathrm{A}\sqrt{{\mathrm{x}}^2 - 12\mathrm{x} +\hspace{0.17em}35}$ + $\log \left|\mathrm{x} - 6 + \sqrt{{\mathrm{x}}^2 - 12\mathrm{x} + 35}\right| + \mathrm{C}$, then A =

• - 1

• $\frac{1}{2}$

• $- \frac{1}{2}$

• 1

${\int }_0^3\left[\mathrm{x}\right]\mathrm{dx} = ...,$ where [x] is greatest integer function

• 3

• 0

• 2

• 1

$\int \frac{{\sec }^8\left(\mathrm{x}\right)}{\csc \left(\mathrm{x}\right)}\mathrm{dx}$ =

• $\frac{{\sec }^8\left(\mathrm{x}\right)}{8} + \mathrm{c}$

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

• $\frac{{\sec }^6\left(\mathrm{x}\right)}{6} + \mathrm{c}$

• $\frac{{\sec }^9\left(\mathrm{x}\right)}{9} + \mathrm{c}$

The tangent to the graph of a continuous function y = f(x) at the point with abscissa x = a forms with the X-axis an angle of $\frac{\mathrm{\pi }}{3}$ and at the point with abscissa x = b an angle of $\frac{\mathrm{\pi }}{4}$, then what the value of integral ${\int }_{\mathrm{a}}^{\mathrm{b}}\left\{\mathrm{f}\text{'}\left(\mathrm{x}\right) + \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)\right\}\mathrm{dx}$

(where f'(x) the derivative off w.r.t. xwhich is assumed to be continuous and similarly f"(x) the double derivative of f w.r.t. x)

• e^b + $\sqrt{3}{\mathrm{e}}^{\mathrm{a}}$

• ${\mathrm{e}}^{\mathrm{b}} - \sqrt{3}{\mathrm{e}}^{\mathrm{a}}$

• ${\mathrm{e}}^{\mathrm{b}} - \sqrt{3{\mathrm{e}}^{\mathrm{a}}}$

• $- {\mathrm{e}}^{\mathrm{b}} - \sqrt{3{\mathrm{e}}^{\mathrm{a}}}$

The point of inflection of the function y = ${\int }_0^{\mathrm{x}}\left({\mathrm{t}}^2 - 3\mathrm{t} + 2\right)\mathrm{dt}$ is

• $\left(\frac{3}{2}, \frac{3}{4}\right)$

• $\left(- \frac{3}{2}, - \frac{3}{4}\right)$

• $\left(- \frac{1}{2}, - \frac{3}{2}\right)$

• $\left(\frac{1}{2}, \frac{3}{2}\right)$

The value of integral $\int \frac{\mathrm{dx}}{\mathrm{x}\sqrt{{\mathrm{x}}^2 - {\mathrm{a}}^2}}$

• $\mathrm{c} - \frac{1}{\mathrm{a}}{\sin }^{-1}\left(\frac{\mathrm{a}}{\left|\mathrm{x}\right|}\right)$

• $\mathrm{c} - \frac{1}{\mathrm{a}}{\cos }^{-1}\left(\frac{\mathrm{a}}{\left|\mathrm{x}\right|}\right)$

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

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