${\int }_0^1\log \left(\frac{1}{\mathrm{x}} - 1\right)d\mathrm{x}$

• 1

• 0

• 2

• None of the above

If [x] denotes the greatest integer less than or equal to x, then the value of the integral ${\int }_0^2{\mathrm{x}}^2\left[\mathrm{x}\right]d\mathrm{x}$  equals

• $\frac{5}{3}$

• $\frac{7}{3}$

• $\frac{8}{3}$

• $\frac{4}{3}$

If $$\begin{gathered} \mathrm{\varphi }\left(\mathrm{t}\right) = \left\{1, \mathrm{for} 0 \le \mathrm{t} < 1 \\ 0, \mathrm{otherwise}\right\}, \mathrm{then} \end{gathered}$$

${\int }_{- 3000}^{3000}\left(\sum _{\mathrm{r}\text{'} = 2014}^{2016}\left(\mathrm{\varphi }\left(\mathrm{t} - \mathrm{r}\text{'}\right)\mathrm{\varphi }\left(\mathrm{t} - 2016\right)\right)\right)\mathrm{dt}$ is

• a real number

• 1

• 0

• does not exist

The value of $\underset{\mathrm{x}\to 2}{\lim }{\int }_2^{\mathrm{x}}\frac{3{\mathrm{t}}^2}{\left(\mathrm{x} - 2\right)}d\mathrm{t}$ is

• 10

• 12

• 18

• 16

Let f(x) denotes the fractional part of a real number x. Then, the value of ${\int }_0^{\sqrt{3}}\mathrm{f}\left({\mathrm{x}}^2\right)d\mathrm{x}$

• $2\sqrt{3} - \sqrt{2} - 1$

• 0

• $\sqrt{2} - \sqrt{3} + 1$

• $\sqrt{3} - \sqrt{2} + 1$

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)

339.

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