${\int }_0^{\mathrm{\pi }}\frac{\mathrm{xdx}}{{\mathrm{a}}^2{\cos }^2\left(\mathrm{x}\right) + {\mathrm{b}}^2{\sin }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{2\mathrm{ab}}$

• $\frac{\mathrm{\pi }}{\mathrm{ab}}$

• $\frac{{\mathrm{\pi }}^2}{2\mathrm{ab}}$

• $\frac{{\mathrm{\pi }}^2}{\mathrm{ab}}$

The differential equation for which sin^-1(x) + sin^-1(y) = c is given by

• $\sqrt{1 - {\mathrm{x}}^2}\mathrm{dy} + \sqrt{1 - {\mathrm{y}}^2}\mathrm{dx} = 0$

• $\sqrt{1 - {\mathrm{x}}^2}\mathrm{dx} + \sqrt{1 - {\mathrm{y}}^2}\mathrm{dy} = 0$

• $\sqrt{1 - {\mathrm{x}}^2}\mathrm{dx} - \sqrt{1 - {\mathrm{y}}^2}\mathrm{dy} = 0$

• $\sqrt{1 - {\mathrm{x}}^2}\mathrm{dy} - \sqrt{1 - {\mathrm{y}}^2}\mathrm{dx} = 0$

23.

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

• ${\mathrm{e}}^{\mathrm{x}}{\sec }^2\left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

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

• ${\mathrm{e}}^{\mathrm{x}}\sec \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{c}$

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

$\int \sqrt{1 + \sin \left(\frac{\mathrm{x}}{4}\right)}\mathrm{dx}$ is equal to

• $8\left(\sin \left(\frac{\mathrm{x}}{8}\right) + \cos \left(\frac{\mathrm{x}}{8}\right)\right) + \mathrm{C}$

• $8\left(\sin \left(\frac{\mathrm{x}}{8}\right) - \cos \left(\frac{\mathrm{x}}{8}\right)\right) + \mathrm{C}$

• $8\left(\cos \left(\frac{\mathrm{x}}{8}\right) - \sin \left(\frac{\mathrm{x}}{8}\right)\right) + \mathrm{C}$

• $\frac{1}{8}\left(\sin \left(\frac{\mathrm{x}}{8}\right) - \cos \left(\frac{\mathrm{x}}{8}\right)\right) + \mathrm{C}$

${\int }_0^{\infty }\frac{\mathrm{xdx}}{\left(1 + \mathrm{x}\right)\left(1 + {\mathrm{x}}^2\right)}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• 0

• 1

• $\frac{\mathrm{\pi }}{4}$

If I_n = $\int {\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}\mathrm{dx}$, then I_n + nI_{n - 1} is equal to

• ${\left(\mathrm{xlog}\left(\mathrm{x}\right)\right)}^{\mathrm{n}}$

• $\mathrm{x}{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}$

• $\mathrm{n}{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}$

• ${\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n} - 1}$

The area included between the parabolas x² = 4y and y² = 4x is (in square units)

• $\frac{4}{3}$

• $\frac{1}{3}$

• $\frac{16}{3}$

• $\frac{8}{3}$