${\int }_0^3\left[\mathrm{x}\right]\mathrm{dx} = ...,$ 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 $\int \frac{\mathrm{dx}}{\mathrm{x}\sqrt{{\mathrm{x}}^2 - {\mathrm{a}}^2}}$

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

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

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

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

The value of ${\int }_0^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.}\frac{{\sin }^{-1}\left(\mathrm{x}\right)}{{\left(1 - {\mathrm{x}}^2\right)}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\mathrm{dx}$ is

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

• $\frac{\mathrm{\pi }}{4} - \frac{1}{2}\log \left(2\right)$

   

• $\frac{\mathrm{\pi }}{4} + \frac{1}{2}\log \left(2\right)$

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

Integral of $\frac{1}{2 + \cos \left(\mathrm{x}\right)}$

• $\frac{1}{\sqrt{3}}{\tan }^{-1}\left(\frac{1}{2}\tan \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{C}$

• $\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{1}{\sqrt{3}}\tan \left(\frac{\mathrm{x}}{2}\right)\right) +\hspace{0.17em}\mathrm{C}$

• $- \sin \left(\mathrm{x}\right)\log \left(2 + \cos \left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\sin \left(\mathrm{x}\right)\log \left(2 + \cos \left(\mathrm{x}\right)\right) + \mathrm{C}$

$\int \frac{\mathrm{dx}}{\sqrt{{\mathrm{x}}^4 + {\mathrm{x}}^6}}$ is equal to

• $- \frac{\sqrt{1 + {\mathrm{x}}^2}}{\mathrm{x}} + \mathrm{C}$

• $\frac{\sqrt{1 + {\mathrm{x}}^2}}{\mathrm{x}} + \mathrm{C}$

• $- \frac{\sqrt{1 - {\mathrm{x}}^2}}{\mathrm{x}} + \mathrm{C}$

• $- \frac{\sqrt{{\mathrm{x}}^2 - 1}}{\mathrm{x}} + \mathrm{C}$

409.

If $\int {\sin }^{-1}\left(\mathrm{x}\right){\cos }^{-1}\left(\mathrm{x}\right)\mathrm{dx} = {\mathrm{f}}^{-1}\left(\mathrm{x}\right)\left[\frac{\mathrm{\pi }}{2}\mathrm{x} - \mathrm{x}{\mathrm{f}}^{-1}\left(\mathrm{x}\right) - 2\sqrt{1 - {\mathrm{x}}^2}\right]$

$\frac{\mathrm{\pi }}{2}\sqrt{1 - {\mathrm{x}}^2} + 2\mathrm{x} +\hspace{0.17em}\mathrm{C}$, then

• f(x) = sin(x)

• f(x) = cos(x)

• f(x) = tan(x)

• None of these

If ${\int }_0^{\mathrm{\pi }}\mathrm{xf}\left({\sin }^2\left(\mathrm{x}\right) + {\sec }^2\left(\mathrm{x}\right)\right)\mathrm{dx} = \mathrm{k}{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{f}\left({\sin }^2\left(\mathrm{x}\right) + {\sec }^2\left(\mathrm{x}\right)\right)\mathrm{dx}$, then the value of k is

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $- \frac{\mathrm{\pi }}{2}$

• None of these