What is ${\int }_{ - 2}^2\mathrm{x}d\mathrm{x} - {\int }_{ - 2}^2\left[\mathrm{x}\right]d\mathrm{x}$

• 0

• 1

• 2

• 4

32.

If ${\int }_{ - 2}^5\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = 4 \mathrm{and} {\int }_0^5\left[1 + \mathrm{f}\left(\mathrm{x}\right)\right]d\mathrm{x} = 7$

then what is ${\int }_{ - 2}^0\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = ?$

•  - 3

• 2

• 3

• 5

What is ${\int }_0^{4\mathrm{\pi }}\left|\cos \left(\mathrm{x}\right)\right|\mathrm{dx}$ = ?

• 0

• 2

• 4

• 8

Let f(x) be a function such that $\mathrm{f}\text{'}\left(\frac{1}{\mathrm{x}}\right) + {\mathrm{x}}^3\mathrm{f}\text{'}\left(\mathrm{x}\right) = 0. \mathrm{What} \mathrm{is} {\int }_{ - 1}^1\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = ?$

• 2f(1)

• 0

• 2f( - 1)

• 4f(1)

What is $\int \frac{{\mathrm{x}}^4 - 1}{{\mathrm{x}}^2\sqrt{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1}}\mathrm{dx} = ?$

• $\sqrt{\frac{{\mathrm{x}}^4 +\hspace{0.17em}{\mathrm{x}}^2 + 1 }{4}} + \mathrm{c}$

• $\sqrt{{\mathrm{x}}^2 +\hspace{0.17em}2 - \frac{1}{{\mathrm{x}}^2}} + \mathrm{c}$

• $\sqrt{{\mathrm{x}}^2 + \frac{1}{{\mathrm{x}}^2} + 1} + \mathrm{c}$

• $\sqrt{\frac{{\mathrm{x}}^4 - {\mathrm{x}}^2 + 1}{\mathrm{x}}} +\hspace{0.17em}\mathrm{c}$

What is $\int {\mathrm{e}}^{\sin \left(\mathrm{x}\right)}\frac{{\mathrm{xcos}}^3\left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}{{\cos }^2\left(\mathrm{x}\right)}\mathrm{dx} = ?$

• $\left(\mathrm{x} + \sec \left(\mathrm{x}\right)\right){\mathrm{e}}^{\sin \left(\mathrm{x}\right)} +\hspace{0.17em}\mathrm{c}$

• $\left(\mathrm{x} - \sec \left(\mathrm{x}\right)\right){\mathrm{e}}^{\sin \left(\mathrm{x}\right)} +\hspace{0.17em}\mathrm{c}$

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

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

If ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\mathrm{dx}}{3\cos \left(\mathrm{x}\right) \hspace{0.17em}5} = {\mathrm{kcot}}^{-1}\left(2\right)$, then what is the value of k ?

• $\frac{1}{4}$

• $\frac{1}{2}$

• 1

• 2

What is ${\int }_1^3\left|1 - {\mathrm{x}}^4\right|d\mathrm{x} = ?$

• $- \frac{232}{5}$

• $- \frac{116}{5}$

• $\frac{116}{5}$

• $\frac{232}{5}$

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{integral} \\ {\mathrm{I}}_1 = {\int }_{ 0}^{\mathrm{\pi }}\frac{\mathrm{x}}{1 + \sin \left(\mathrm{x}\right)}d\mathrm{x} \mathrm{and} \\ {\mathrm{I}}_2 = {\int }_{ 0}^{\mathrm{\pi }}\frac{\left(\mathrm{\pi } - \mathrm{x}\right)\mathrm{dx}}{1 - \sin \left(\mathrm{\pi } +\hspace{0.17em}\mathrm{x}\right)} \end{gathered}$$

What is the value of I₁ + I₂ ?

• 2$\mathrm{\pi }$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• 0

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{integral} \\ {\mathrm{I}}_1 = {\int }_0^{\mathrm{\pi }}\frac{\mathrm{x}}{1 + \sin \left(\mathrm{x}\right)}d\mathrm{x} \mathrm{and} {\mathrm{I}}_2 = {\int }_0^{\mathrm{\pi }}\frac{\left(\mathrm{\pi } - \mathrm{x}\right)\mathrm{dx}}{1 - \sin \left(\mathrm{\pi } +\hspace{0.17em}\mathrm{x}\right)} \end{gathered}$$

What is the value of I₁

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• 2$\mathrm{\pi }$