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

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

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

• ${\tan }^{-1}\left({\sin }^2\left(\mathrm{x}\right)\right) + \mathrm{C}$

• ${\tan }^{-1}\left(2{\sin }^2\left(\mathrm{x}\right)\right) + \mathrm{C}$

$\int {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)}\left(1 + \frac{\mathrm{x}}{1 +\hspace{0.17em}{\mathrm{x}}^2}\right)\mathrm{dx}$ is equal to

• ${\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{x}\right)} +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)} +\hspace{0.17em}\mathrm{c}$

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

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

493.

$\int \csc \left(\mathrm{x} - \mathrm{a}\right)\csc \left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{- 1}{\sin \left(\mathrm{a}\right)}\log \left|\sin \left(\mathrm{x}\right)\csc \left(\mathrm{x} - \mathrm{a}\right)\right| +\hspace{0.17em}\mathrm{c}$

• $\frac{- 1}{\sin \left(\mathrm{a}\right)}\log \left[\sin \left(\mathrm{x} - \mathrm{a}\right)\sin \left(\mathrm{x}\right)\right] +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{\sin \left(\mathrm{a}\right)}\log \left[\sin \left(\mathrm{x}\right)\csc \left(\mathrm{x} - \mathrm{a}\right)\right] +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{\sin \left(\mathrm{a}\right)}\log \left[\sin \left(\mathrm{x} - \mathrm{a}\right)\sin \left(x\right)\right] +\hspace{0.17em}\mathrm{c}$

If f(x) = ${\int }_{- 1}^{\mathrm{x}}\left|\mathrm{t}\right|\mathrm{dt}$, then for any x $\ge$ 0, f(x) is equal to

• 1 - x²

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

• 1 + x²

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

${\int }_1^3\frac{\sqrt{4 - \mathrm{x}}}{\sqrt{\mathrm{x}} + \sqrt{4 - \mathrm{x}}}\mathrm{dx}$ is equal to

• 1

• 3

• 2

• 0

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

• $- \frac{2}{35}$

• $\frac{4}{35}$

• $\frac{24}{35}$

• $- \frac{8}{35}$