${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\log \left(1 + \tan \left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{8}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{\mathrm{\pi }}{4}{\log }_2\left(\mathrm{e}\right)$

• $\frac{\mathrm{\pi }}{4}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{\mathrm{\pi }}{8}{\log }_{\mathrm{e}}\left(\frac{1}{2}\right)$

$\int \frac{\left({\mathrm{x}}^3 +\hspace{0.17em}3{\mathrm{x}}^2 + 3\mathrm{x} + 1\right)}{{\left(\mathrm{x} + 1\right)}^5}\mathrm{dx}$ is equal to

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

• $\frac{1}{5}\log \left(\mathrm{x} +\hspace{0.17em}1\right) + \mathrm{c}$

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

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

$\int \frac{\csc \left(\mathrm{x}\right)}{{\cos }^2\left(1 + \log \left(\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right)\right)\right)}\mathrm{dx}$ is equal to :

• ${\sin }^2\left(1 + \log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)\right) + \mathrm{c}$

• $\tan \left(1 + \log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)\right) + \mathrm{c}$

• ${\sec }^2\left(1 + \log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)\right) + \mathrm{c}$

• $- \tan \left(1 + \log \left(\tan \left(\frac{\mathrm{x}}{2}\right)\right)\right) + \mathrm{c}$

$\int \frac{\mathrm{dx}}{\mathrm{x}\sqrt{{\mathrm{x}}^6 - 16}}$ is equa to :

• $\frac{1}{3}{\sec }^{-1}\left(\frac{{\mathrm{x}}^3}{4}\right) +\hspace{0.17em}\mathrm{c}$

• ${\cosh }^{-1}\left(\frac{{\mathrm{x}}^3}{4}\right) +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{12}{\sec }^{-1}\left(\frac{{\mathrm{x}}^3}{4}\right) +\hspace{0.17em}\mathrm{c}$

• ${\sec }^{-1}\left(\frac{{\mathrm{x}}^3}{4}\right) +\hspace{0.17em}\mathrm{c}$

If I₁ = ${\mathrm{I}}_1 = {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(\mathrm{x}\right)\mathrm{dx} \mathrm{and} {\mathrm{I}}_2 = {\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{xcos}\left(\mathrm{x}\right)\mathrm{dx}$, then which one of the following is true ?

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

• ${\mathrm{I}}_2 - {\mathrm{I}}_1= \frac{\mathrm{\pi }}{2}$

• ${\mathrm{I}}_1 + {\mathrm{I}}_{2 }= 0$

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

If f(x) is defined [- 2, 2] by f(x) = 4 - 3x + 1 and g(x) = $\frac{\mathrm{f}\left(- \mathrm{x}\right) - \left(\mathrm{f}\left(\mathrm{x}\right)\right)}{{\mathrm{x}}^2 + 3}$, then ${\int }_{- 2}^2\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to :

• 64

• - 48

• 0

• 24

767.

The value of the integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left({\sin }^{100}\left(\mathrm{x}\right) - {\cos }^{100}\left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• $\frac{1}{100}$

• $\frac{100!}{{\left(100\right)}^{100}}$

• $\frac{\mathrm{\pi }}{100}$

• 0

If k${\int }_0^1\mathrm{x} . \mathrm{f}\left(3\mathrm{x}\right)\mathrm{dx} = {\int }_0^3\mathrm{t} . \mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$, then the value of k is

• 9

• 3

• $\frac{1}{9}$

• $\frac{1}{3}$

The value of $\int \frac{1}{1 + \cos \left(8\mathrm{x}\right)}\mathrm{dx}$ is

• $\frac{\tan \left(2\mathrm{x}\right)}{8} + \mathrm{c}$

• $\frac{\tan \left(8\mathrm{x}\right)}{8} + \mathrm{c}$

• $\frac{\tan \left(4\mathrm{x}\right)}{4} + \mathrm{c}$

• $\frac{\tan \left(4\mathrm{x}\right)}{8} + \mathrm{c}$

The value of $\int {\mathrm{e}}^{\mathrm{x}}\left({\mathrm{x}}^5 + 5{\mathrm{x}}^4 + 1\right)\mathrm{dx}$ is

• e^x . x⁵ + c

• e^x . x⁵ + e^x + c

• e^{x + 1} . x⁵ + c

• 5x⁴ . e^x + c