51.

The value of $\int \frac{\left(\mathrm{x} - 2\right)}{{\left\{{\left(\mathrm{x} - 2\right)}^2 {\left(\mathrm{x} + 3\right)}^7\right\}}^{1/3}}\mathrm{dx}$

• $\frac{3}{20}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} +\hspace{0.17em}3}\right)}^{4/3} + \mathrm{C}$

• $\frac{3}{20}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} +\hspace{0.17em}3}\right)}^{3/4} + \mathrm{C}$

• $\frac{5}{12}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} +\hspace{0.17em}3}\right)}^{4/3} + \mathrm{C}$

• $\frac{3}{20}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} +\hspace{0.17em}3}\right)}^{5/3} + \mathrm{C}$

If f(x) = $$\begin{gathered} \left\{2{\mathrm{x}}^2 + 1, \mathrm{x} \le 1 \\ 4{\mathrm{x}}^3 - 1, \mathrm{x} > 1\right\} \end{gathered}$$, then ${\int }_0^2\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• 47/3

• 50/3

• 1/3

• 47/2

If $\mathrm{I} = {\int }_0^2{\mathrm{e}}^{{\mathrm{x}}^4}\left(\mathrm{x} - \mathrm{\alpha }\right)d\mathrm{x} = 0$, then $\mathrm{\alpha }$ lies in the interval

• (0, 2)

• (- 1, 0)

• (2, 3)

• (- 2, - 1)

The value of $\underset{\mathrm{x}\to 0}{\lim }\frac{{\int }_0^{{\mathrm{x}}^2}\cos \left({\mathrm{t}}^2\right)d\mathrm{x}}{\mathrm{xsin}\left(\mathrm{x}\right)}$

• 1

• - 1

• 2

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

Let f(x) = $\max \left\{\mathrm{x} +\hspace{0.17em}\left|\mathrm{x}\right|, \mathrm{x} - \left[\mathrm{x}\right]\right\}$, where [x] denotes the greatest integer $\le \mathrm{x}$. Then, the values of ${\int }_{- 3}^3\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• 0

• 51/2

• 21/2

• 1

Suppose $\mathrm{M} = {\int }_0^{\mathrm{\pi }/2}\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x} + 2}d\mathrm{x}, \mathrm{N} = {\int }_0^{\mathrm{\pi }/4}\frac{\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}{{\left(\mathrm{x} + 1\right)}^2}d\mathrm{x}$. Then, the values of (M - N) equals

• $\frac{3}{\mathrm{\pi } + 2}$

• $\frac{2}{\mathrm{\pi } - 4}$

• $\frac{4}{\mathrm{\pi } - 2}$

• $\frac{2}{\mathrm{\pi } + 4}$

The value of the integral

${\int }_{- 1}^1\left\{\frac{{\mathrm{x}}^{2013}}{{\mathrm{e}}^{\left|\mathrm{x}\right|}\left({\mathrm{x}}^2 +\hspace{0.17em}\cos \left(\mathrm{x}\right)\right)} + \frac{1}{{\mathrm{e}}^{\left|\mathrm{x}\right|}}\right\}d\mathrm{x}$ is equal to

• 0

• $1 - {\mathrm{e}}^{- 1}$

• $2{\mathrm{e}}^{- 1}$

• $2\left(1 - {\mathrm{e}}^{- 1}\right)$

The value of I = ${\int }_0^{\mathrm{\pi }/4}\left({\tan }^{\mathrm{n} + 1}\left(\mathrm{x}\right)\right)d\mathrm{x} + \frac{1}{2}{\int }_0^{\mathrm{\pi }/2}\left({\tan }^{\mathrm{n} + 1}\left(\frac{\mathrm{x}}{2}\right)\right)d\mathrm{x}$ is

• $\frac{1}{\mathrm{n}}$

• $\frac{\mathrm{n} + 2}{2\mathrm{n} + 1}$

• $\frac{2\mathrm{n} - 1}{\mathrm{n}}$

• $\frac{2\mathrm{n} - 3}{3\mathrm{n} - 2}$

The value of the integral

${\int }_1^2{\mathrm{e}}^{\mathrm{x}}\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right) + \frac{\mathrm{x} + 1}{\mathrm{x}}\right)d\mathrm{x}$

• ${\mathrm{e}}^2\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

• ${\mathrm{e}}^2 - \mathrm{e}$

• ${\mathrm{e}}^2\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right) - \mathrm{e}$

• ${\mathrm{e}}^2 - \mathrm{e}\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

If [a] denote the greatest integer which is less than or equal to a. Then, the value of the integral ${\int }_{- \frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left[\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)\right]d\mathrm{x}$ is

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $- \mathrm{\pi }$

• $- \frac{\mathrm{\pi }}{2}$