${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(2\mathrm{x}\right) . \log \left(\tan \left(\mathrm{x}\right)\right)\mathrm{dx}$

• 0

• 2

• 4

• 7

If $∆\mathrm{x} = \left|\begin{array}{ccc}1& \cos \left(\mathrm{x}\right)& 1 - \cos \left(\mathrm{x}\right)\\ 1 + \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& 1 + \sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\\ \sin \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)& 1\end{array}\right|$, then ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}∆\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{1}{4}$

• $\frac{1}{2}$

• 0

• $- \frac{1}{4}$

${\int }_0^{2\mathrm{\pi }}\left(\sin \left(\mathrm{x}\right) + \left|\sin \left(\mathrm{x}\right)\right|\right)\mathrm{dx}$ is equal to

• 0

• 4

• 8

• 1

The value of ${\int }_0^{\sqrt{2}}\left[{\mathrm{x}}^2\right]\mathrm{dx}$, where [.] is the greatest integer function, is

• 2 - $\sqrt{2}$

• 2 + $\sqrt{2}$

• $\sqrt{2}$ - 1

• $\sqrt{2}$ - 2

If l (m,n) = ${\int }_0^1{\mathrm{t}}^{\mathrm{m}}{\left(1 + \mathrm{t}\right)}^{\mathrm{n}}\mathrm{dt}$, then the expression for l (m, n) in terms of l (m + 1, n + 1) is

• $\frac{2^{\mathrm{n}}}{\mathrm{m} + 1} - \frac{\mathrm{n}}{\mathrm{m} + 1} . \mathrm{l} \left(\mathrm{m} + 1, \mathrm{n} - 1\right)$

• $\frac{\mathrm{n}}{\mathrm{m} + 1} . \mathrm{l} \left(\mathrm{m} + 1, \mathrm{n} - 1\right)$

• $\frac{2\mathrm{n}}{\mathrm{m} + 1} + \frac{\mathrm{n}}{\mathrm{m} + 1} . \mathrm{l} \left(\mathrm{m} + 1, \mathrm{n} - 1\right)$

• $\frac{\mathrm{m}}{\mathrm{n} + 1} . \mathrm{l} \left(\mathrm{m} + 1, \mathrm{n} - 1\right)$

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

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

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

• ${\mathrm{xe}}^{\mathrm{x} + {\mathrm{x}}^{- 1}} + \mathrm{C}$

• ${\mathrm{xe}}^{\mathrm{x} + {\mathrm{x}}^{- 1}} \mathrm{x} + \mathrm{C}$

If f(x) = x - [x], for every real number x, where [x] is the integral part of x. Then ${\int }_{- 1}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• 1

• 2

• 0

• $\frac{1}{2}$

The value of integral

${\int }_{- 1}^1{\left[{\left(\frac{\mathrm{x} + 1}{\mathrm{x} - 1}\right)}^2 + {\left(\frac{\mathrm{x} + 1}{\mathrm{x} - 1}\right)}^2 - 2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{dx}$ is

• $\log \left(\frac{4}{3}\right)$

• $4\log \left(\frac{3}{4}\right)$

• $4\log \left(\frac{4}{3}\right)$

• $\log \left(\frac{3}{4}\right)$

$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + \sqrt{2}}$ equals to

• $- \frac{1}{\sqrt{2}}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $\frac{1}{2}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $\frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $- \frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

The value of I = ${\int }_0^1\mathrm{x}\left|\mathrm{x} - \frac{1}{2}\right|\mathrm{dx}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• None of these