781.

If ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = 5, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} ... + {\int }_0^1{\mathrm{x}}^9\mathrm{f}\left({\mathrm{x}}^{10}\right)\mathrm{dx}$ is equal to

• 125

• 625

• 275

• 55

If $\int \mathrm{f}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right) . \cos \left(\mathrm{x}\right)\mathrm{dx} = \frac{1}{2\left({\mathrm{b}}^2 - {\mathrm{a}}^2\right)}\log \left(\mathrm{f}\left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{c}$, where c is the constant of integration, then f(x) is

• $\frac{2}{\mathrm{abcos}\left(2\mathrm{x}\right)}$

• $\frac{2}{\left({\mathrm{b}}^2 - {\mathrm{a}}^2\right)\cos \left(2\mathrm{x}\right)}$

• $\frac{2}{\mathrm{ab}\sin \left(2\mathrm{x}\right)}$

• $\frac{2}{\left({\mathrm{b}}^2 - {\mathrm{a}}^2\right)\sin \left(2\mathrm{x}\right)}$

If $\int \frac{\sqrt{\mathrm{x}}}{\mathrm{x}\left(\mathrm{x} + 1\right)}\mathrm{dx} = {\mathrm{ktan}}^{-1}\left(\mathrm{m}\right)$, then (k, m) is

• (2, x)

• (1, x)

• $\left(1, \sqrt{\mathrm{x}}\right)$

• $\left(2, \sqrt{\mathrm{x}}\right)$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{3 + \sin \left(2\mathrm{x}\right)}\mathrm{dx}$ is

• $\frac{1}{4}\log \left(3\right)$

• $\log \left(3\right)$

• $\frac{1}{2\log \left(3\right)}$

• $2\log \left(3\right)$

${\int }_0^1\mathrm{x}{\left(1 - \mathrm{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{dx}$ is

• $- \frac{2}{35}$

• $\frac{4}{35}$

• $\frac{24}{35}$

• $- \frac{8}{35}$

The value of ${\int }_0^4\left|\mathrm{x} - 1\right|\mathrm{dx}$ is

• $\frac{5}{2}$

• 5

• 4

• 1

If I_n = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$, where where n is apositive integer, then ${\mathrm{I}}_{10} + {\mathrm{I}}_8$ is

• $\frac{1}{9}$

• $\frac{1}{8}$

• $\frac{1}{7}$

• 9

If I_n = $\int {\mathrm{e}}^{\mathrm{x}}\left[\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{1 - {\sin }^2\left(\mathrm{x}\right)}\right]\mathrm{dx}$ is

• $\left({\mathrm{e}}^{\mathrm{x}} . \csc \left(\mathrm{x}\right)\right) +\hspace{0.17em}\mathrm{C}$

• $\left({\mathrm{e}}^{\mathrm{x}} . \cot \left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\left({\mathrm{e}}^{\mathrm{x}} . \sec \left(\mathrm{x}\right)\right) + \mathrm{C}$

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

When x > 0, then $\int {\cos }^{-1}\left(\frac{1 - {\mathrm{x}}^2}{1 +\hspace{0.17em}{\mathrm{x}}^2}\right)\mathrm{dx}$ is

• $2\left[{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right) - \log \left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)\right] + \mathrm{C}$

• $2\left[{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right) + \log \left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)\right] + \mathrm{C}$

• $2{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right) + \log \left(1 +\hspace{0.17em}{\mathrm{x}}^2\right) + \mathrm{C}$

• $2{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right) - \log \left(1 +\hspace{0.17em}{\mathrm{x}}^2\right) + \mathrm{C}$

If the area between y = mx² and x = my² (m > 0) is 1/4 sq units, then the value of m is

• $\pm 3\sqrt{2}$

• $\pm \frac{2}{\sqrt{3}}$

• $\sqrt{2}$

• $\sqrt{3}$