${\int }_{\raisebox{1ex}{$\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

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

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

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

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

If I₁ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{xsin}\left(\mathrm{x}\right)\mathrm{dx}$, I₂ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{xcos}\left(\mathrm{x}\right)\mathrm{dx}$, then whi ch one f the followin is true ?

• I₁ = I₂

• I₁ + I₂ = 0

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

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

The value of ${\int }_{- 1}^2\frac{\left|\mathrm{x}\right|}{\mathrm{x}}\mathrm{dx}$, is

• 0

• 1

• 2

• 3

${\int }_0^{\mathrm{\pi }}\frac{{\cos }^4\left(\mathrm{x}\right)}{{\cos }^4\left(\mathrm{x}\right) + {\sin }^4\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{8}$

• $\mathrm{\pi }$

If f(x) = $\mathrm{f}\left(\mathrm{\pi } + \mathrm{e} - \mathrm{x}\right) \mathrm{and} {\int }_{\mathrm{e}}^{\mathrm{\pi }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \frac{2}{\mathrm{e} +\hspace{0.17em}\mathrm{\pi }}, \mathrm{then} {\int }_{\mathrm{e}}^{\mathrm{\pi }}\mathrm{xf}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\mathrm{\pi } - \mathrm{e}$

• $\frac{\mathrm{\pi } + \mathrm{e}}{2}$

• 1

• $\frac{\mathrm{\pi } - \mathrm{e}}{2}$

800.

If linear function f(x) and g(x) satisfy $\int \left[\left(3\mathrm{x} - 1\right)\cos \left(\mathrm{x}\right) + \left(1 - 2\mathrm{x}\right)\sin \left(\mathrm{x}\right)\right]\mathrm{dx}$ = $\mathrm{f}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right) + \mathrm{g}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right) + \mathrm{C}$, then

• f(x) = 3(x - 1)

• f(x) = 3x - 5

• g(x) = 3(x - 1)

• g(x) = 3 + x