$\int {\mathrm{e}}^{\mathrm{xlog}\left(\mathrm{a}\right)}{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to :

• $\frac{{\mathrm{a}}^{\mathrm{x}}}{\log \left(\mathrm{ae}\right)} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{1 + {\log }_{\mathrm{e}}\left(\mathrm{a}\right)} + \mathrm{c}$

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

• $\frac{{\displaystyle {\left(\mathrm{ae}\right)}^{\mathrm{x}}}}{{\displaystyle {\log }_e\left(\mathrm{ae}\right)}} + \mathrm{c}$

$\int \sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\mathrm{dx}$ is equal to :

• $2\left[\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1} - {\tan }^{-1}\left(\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\right)\right] + \mathrm{c}$

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

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

• $2\left[\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1} + {\tan }^{-1}\left(\sqrt{{\mathrm{e}}^{\mathrm{x}} - 1}\right)\right] + \mathrm{c}$

293.

If I₁ = $\int {\sin }^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ and I₂ = $\int {\sin }^{-1}\left(\sqrt{1 - {\mathrm{x}}^2}\right)\mathrm{dx}$ then : 

• I₁ = I₂

• ${\mathrm{I}}_{2 }= \frac{\mathrm{\pi }}{2}{\mathrm{I}}_1$

• I₁ + I₂ = $\frac{\mathrm{\pi }}{2}\mathrm{x}$

• I₁ + I₂ = $\frac{\mathrm{\pi }}{2}$

$\int {\cos }^{\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right){\sin }^{\raisebox{1ex}{$-11$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to:

• $\log \left|{\sin }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\right| + \mathrm{c}$

• $\frac{4}{7}{\tan }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right) + \mathrm{c}$

• $\frac{- 7}{4}{\tan }^{\raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right) + \mathrm{c}$

• $\log \left|{\cos }^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\right| + \mathrm{c}$

$\int \frac{\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right)}{\sqrt{\sin \left(2\mathrm{\theta }\right)}}\mathrm{d\theta }$ is equal to :

• $\log \left|\cos \left(\mathrm{\theta }\right) - \sin \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right|$

• $\log \left|\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right|$

• ${\sin }^{-1}\left(\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right)\right)$

• ${\sin }^{-1}\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right)$

${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{\mathrm{dx}}{1 + \sqrt{\tan \left(\mathrm{x}\right)}}$ is equal to :

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

• $2\mathrm{\pi }$

${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}\frac{{\sin }^4\left(\mathrm{x}\right)}{{\sin }^4\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)}$dx is equal to :

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

• $2\mathrm{\pi }$

The value of $\frac{2^{\sin \left(\mathrm{x}\right)}}{2^{\sin \left(\mathrm{x}\right)} + 2^{\cos \left(\mathrm{x}\right)}}\mathrm{dx}$ is :

• 2

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

• $2\mathrm{\pi }$

If f is continuous function, then :

• ${\int }_{- 2}^2\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_0^2\left[\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(- \mathrm{x}\right)\right]\mathrm{dx}$

• ${\int }_{- 3}^{5}2\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = {\int }_{- 6}^{10}\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

• ${\int }_{- 3}^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 4}^4\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

• ${\int }_{- 3}^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 2}^6\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

If $\int \frac{\sqrt{\mathrm{x}}}{\mathrm{x} + 1}\mathrm{dx} = \mathrm{A}\sqrt{\mathrm{x}} + {\mathrm{Btan}}^{-1}\left(\sqrt{\mathrm{x}}\right) + \mathrm{c}$, then :

• A = 1, B = 1

• A = 1, B = 2

• A = 2, B = 2

• A = 2, B = - 2