501.

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

• $\pm 3\sqrt{2}$

• $\pm \frac{2}{\sqrt{3}}$

• $\sqrt{2}$

• $\sqrt{3}$

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

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{6}$

If [x] is the greatest integer function not greater than x, then  ${\int }_0^{11}\left[\mathrm{x}\right]\mathrm{dx}$ is equal to

• 45

• 66

• 35

• 55

If $\mathrm{n} \in \mathrm{N}$ and I_n = $\int {\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}\mathrm{dx}$, then I_n + nI_{n - 1} is equal to

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

• $\mathrm{x}{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}} +\hspace{0.17em}\mathrm{C}$

• ${\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n} - 1}$

• $\frac{{\left(\log \left(\mathrm{x}\right)\right)}^{\mathrm{n}}}{\mathrm{n}}$

$\int \frac{{\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)}{{\sin }^{\mathrm{n} + 1}\left(\mathrm{x}\right)}\mathrm{dx} \left(\mathrm{where}, \mathrm{n} \ne 0\right)$ is equal to

• $\frac{{\cot }^{\mathrm{n}}\left(\mathrm{x}\right)}{\mathrm{n}} + \mathrm{C}$

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

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

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

$\int \frac{\left(\mathrm{x} - 1\right){\mathrm{e}}^{\mathrm{x}}}{{\left(\mathrm{x} +\hspace{0.17em}1\right)}^3}\mathrm{dx}$ is equal to

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

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

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

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