$\int {\left(27{\mathrm{e}}^{9\mathrm{x}} + {\mathrm{e}}^{12\mathrm{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{dx}$ is equal to

• $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right){\left(27 + {\mathrm{e}}^{3\mathrm{x}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + \mathrm{C}$

• $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right){\left(27 + {\mathrm{e}}^{3\mathrm{x}}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + \mathrm{C}$

• $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right){\left(27 + {\mathrm{e}}^{3\mathrm{x}}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + \mathrm{C}$

• $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right){\left(27 + {\mathrm{e}}^{3\mathrm{x}}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + \mathrm{C}$

   

$\int \frac{4\mathrm{dx}}{{\mathrm{x}}^2\sqrt{4 - 9{\mathrm{x}}^2}}$ is equal to

• $\sqrt{4 - 9{\mathrm{x}}^2} + \mathrm{C}$

• $\frac{- 2}{3}\sqrt{4 - 9{\mathrm{x}}^2} + \mathrm{C}$

• $\frac{- \sqrt{4 - 9{\mathrm{x}}^2}}{\mathrm{x}} + \mathrm{C}$

• $\frac{2}{3}\frac{\sqrt{4 - 9{\mathrm{x}}^2}}{\mathrm{x}} + \mathrm{C}$

$\int {\mathrm{e}}^{- \mathrm{x}}\csc \left(\mathrm{x}\right)\left(1 + \cot \left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{- \mathrm{x}}\csc \left(\mathrm{x}\right) + \mathrm{C}$

• $- {\mathrm{e}}^{- \mathrm{x}}\csc \left(\mathrm{x}\right) + \mathrm{C}$

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

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

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

• $\log \left|1 - {\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $\log \left|1 + {\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $\log \left|1 + {\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $\log \left|1 - {\mathrm{e}}^{- \mathrm{x}}\sin \left(\mathrm{x}\right)\right| + \mathrm{C}$

The solution of ${\int }_{\sqrt{2}}^{\mathrm{x}}\frac{\mathrm{dt}}{\sqrt{{\mathrm{t}}^2 - 1}} = \frac{\mathrm{\pi }}{12}$ is

• 1

• 2

• 3

• 4

Let f(x) = x² - 2. If ${\int }_3^6\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = 3\mathrm{f}\left(\mathrm{c}\right)$ for some c $\in$ (3, 6),  then the value of c is equal to

• $\sqrt{12}$

• $\sqrt{21}$

• $\sqrt{19}$

• $\sqrt{17}$

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

• $\frac{1}{2}\log \left(2\right)$

• $\log \left(2\right)$

• $\frac{1}{2}\log \left(3\right)$

• $\log \left(3\right)$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\mathrm{dx}}{1 +\hspace{0.17em}{\tan }^3\left(\mathrm{x}\right)}$ is equal to

• 1

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

219.

If f(x) = ${\int }_1^{\mathrm{x}}{\sin }^2\left(\frac{\mathrm{t}}{2}\right)\mathrm{dt}$, then the value of $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{\pi } +\hspace{0.17em}\mathrm{x}\right) - \mathrm{f}\left(\mathrm{\pi }\right)}{\mathrm{x}}$ is equal to

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{3}{4}$

• 1

$\int \frac{1}{{\mathrm{x}}^2{\left({\mathrm{x}}^4 + 1\right)}^{{\displaystyle \frac{3}{4}}}}\mathrm{dx}$ is equal to

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{3}{4}}}}{\mathrm{x}} + \mathrm{C}$

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{1}{4}}}}{2\mathrm{x}} + \mathrm{C}$

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{1}{4}}}}{\mathrm{x}} + \mathrm{C}$

• $- \frac{{\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{1}{4}}}}{{\mathrm{x}}^2} + \mathrm{C}$