41.

If five dices are tossed, then what is the probability that the five numbers shown will be different?

• $\frac{5}{54}$

• $\frac{5}{18}$

• $\frac{5}{27}$

• $\frac{5}{81}$

If the events A and B are independent and if $\mathrm{P}\left(\overline{\mathrm{A}}\right) = \frac{2}{3}, \mathrm{P}\left(\overline{\mathrm{B}}\right) = \frac{2}{7}$, then $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right)$ is equal to

• $\frac{4}{21}$

• $\frac{3}{21}$

• $\frac{5}{21}$

• $\frac{2}{21}$

$\int \frac{1}{\mathrm{x}\left(\log \left(\mathrm{x}\right)\right)\log \left(\log \left(\mathrm{x}\right)\right)}\mathrm{dx}$ is equal to

• $\log \left(\log \left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\log \left|\log \left(\mathrm{xlog}\left(\mathrm{x}\right)\right)\right| + \mathrm{C}$

• $\log \left|\log \left|\log \left(\log \left(\mathrm{x}\right)\right)\right|\right| + \mathrm{C}$

• $\log \left|\log \left(\log \left(\mathrm{x}\right)\right)\right| + \mathrm{C}$

$\int \frac{3^{\mathrm{x}}}{\sqrt{1 - 9^{\mathrm{x}}}}$dx is equal to

• $\log \left(3\right){\sin }^{-1}\left(3^{\mathrm{x}}\right) + \mathrm{C}$

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

• $\left(\frac{1}{3}\right){\sin }^{-1}\left(3^{\mathrm{x}}\right) +\hspace{0.17em}\mathrm{C}$

• $\frac{1}{9}{\sin }^{-1}\left(3^{\mathrm{x}}\right) +\hspace{0.17em}\mathrm{C}$

The value of the integral $\int \frac{\cos \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

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

• $\frac{1}{2}\left[\mathrm{x} + \log \left|\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right|\right] +\hspace{0.17em}\mathrm{C}$

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

• $\frac{\mathrm{x}}{2} + \log \left|\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right| + \mathrm{C}$

$\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

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