If f(x) = ${\int }_{- 1}^{\mathrm{x}}\left|\mathrm{t}\right|d\mathrm{t}$, then for any $\mathrm{x} \ge 0$, f(x) is equal to

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

• $\left(1 - {\mathrm{x}}^2\right)$

• $\frac{1}{2}\left(1 + {\mathrm{x}}^2\right)$

• $\left(1 + {\mathrm{x}}^2\right)$

Let I = ${\int }_0^{100\mathrm{\pi }}\sqrt{\left(1 - \cos \left(2\mathrm{x}\right)\right)}d\mathrm{x}$, then

• I = 0

• I = 200$\sqrt{2}$

• I = $\mathrm{\pi }\sqrt{2}$

• I = 100

Let f be a non-constant continuous function for all x > 0. Let f satisfy the relation f(ax) f(a-x)=1 for some a $\in$ R*. Then, I = ${\int }_0^{\mathrm{a}}\frac{\mathrm{dx}}{1 + \mathrm{f}\left(\mathrm{x}\right)}$ is equal to

• a

• $\frac{\mathrm{a}}{4}$

• $\frac{\mathrm{a}}{2}$

• f(a)

$\int \frac{\log \left(\sqrt{\mathrm{x}}\right)}{3\mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{1}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{C}$

• $\frac{2}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{C}$

• $\frac{2}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{C}$

• $\frac{1}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{C}$

45.

$\int 2^{\mathrm{x}}\left[\mathrm{f}\text{'}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{x}\right)\log \left(2\right)\right]\mathrm{dx}$

• 2^xf'(x) + C

• 2^xlog(2) + C

• 2^xf(x) + C

• 2^x + C

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

• 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

• $\sqrt{2} - \sqrt{3} + 1$

• $\sqrt{3} - \sqrt{2} + 1$