The value of f(x) is given only at x = 0, $\frac{1}{3}, \frac{2}{3}$ wich of the following can be used to evaluate ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ approximately

• Trapezoidal rule

• Simpson's rule

• Trapezoidal as well as Simpson's rule

• None of the above

If f(x) = $\frac{{\mathrm{e}}^{\mathrm{x}}}{1 + {\mathrm{e}}^{\mathrm{x}}}, {\mathrm{I}}_1 = {\int }_{\mathrm{f}\left(- \mathrm{a}\right)}^{\mathrm{f}\left(\mathrm{a}\right)}\mathrm{xg}\left\{\mathrm{x}\left(1 - \mathrm{x}\right)\right\}\mathrm{dx}$ and ${\mathrm{I}}_2 = {\int }_{\mathrm{f}\left(- \mathrm{a}\right)}^{\mathrm{f}\left(\mathrm{a}\right)}\mathrm{g}\left\{\mathrm{x}\left(1 - \mathrm{x}\right)\right\}\mathrm{dx}$, the value of $\frac{{\mathrm{I}}_2}{{\mathrm{I}}_1}$ is

• 2

• - 3

• - 1

• 1

${\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• ${\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{a} - \mathrm{x}\right)\mathrm{dx}$

• ${\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x} - \mathrm{a}\right)\mathrm{dx}$

• ${\int }_0^{\mathrm{a}}\mathrm{f}\left(2\mathrm{a} - \mathrm{x}\right)\mathrm{dx}$

• ${\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x} + 2\mathrm{a}\right)\mathrm{dx}$

$\int {\sin }^4\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{1}{8}\left[3\mathrm{x} + \frac{\sin \left(4\mathrm{x}\right)}{4} - \frac{4\sin \left(2\mathrm{x}\right)}{2}\right] + \mathrm{c}$

• $\frac{1}{8}\left[3\mathrm{x} + \frac{\sin \left(4\mathrm{x}\right)}{4} + \frac{4\sin \left(2\mathrm{x}\right)}{2}\right] + \mathrm{c}$

• $\frac{1}{4}\left[3\mathrm{x} + \frac{\sin \left(4\mathrm{x}\right)}{4} - \frac{4\cos \left(2\mathrm{x}\right)}{2}\right] + \mathrm{c}$

• $\frac{1}{8}\left[3\mathrm{x} + \frac{\sin \left(4\mathrm{x}\right)}{4} + \frac{4\cos \left(4\mathrm{x}\right)}{2}\right] + \mathrm{c}$

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

• $\mathrm{x} + \mathrm{xlog}\left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{xlog}\left(\mathrm{x}\right) - \mathrm{x} +\hspace{0.17em}\mathrm{c}$

• ${\mathrm{x}}^2\log \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\frac{1}{\mathrm{x}}\log \left(\mathrm{x}\right) + \mathrm{xc}$

Intersection point of f₁(x) = ${\int }_2^{\mathrm{x}}\left(2\mathrm{t} - 5\right)\mathrm{dt} \mathrm{and} {\mathrm{f}}_2\left(\mathrm{x}\right) = {\int }_0^{\mathrm{x}}2\mathrm{tdt}$ is

• $\left(\frac{6}{5}, \frac{36}{25}\right)$

• $\left(\frac{2}{3}, \frac{4}{9}\right)$

• $\left(\frac{1}{3}, \frac{1}{9}\right)$

• $\left(\frac{1}{5}, \frac{1}{25}\right)$

557.

The value of $\underset{\mathrm{n}\to \infty }{\lim }\sum _{\mathrm{r} = 1}^{\mathrm{n}}\frac{1}{\mathrm{n}}{\mathrm{e}}^{\raisebox{1ex}{$\mathrm{r}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n}$}\right.}$ is

• e

• e - 1

• 1 - e

• 1 + e

${\int }_0^2\sqrt{\frac{2 + \mathrm{x}}{2 - \mathrm{x}}}\mathrm{dx}$ is equal to

• $\mathrm{\pi } + 2$

• $\mathrm{\pi } + \frac{3}{2}$

• $\mathrm{\pi } + 1$

• None of these

$\int {\mathrm{e}}^{\mathrm{x}}\sin \left({\mathrm{e}}^{\mathrm{x}}\right)\mathrm{dx}$ is equal to

• $- \cos \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{c}$

• $\cos \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{c}$

• $- \csc \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{c}$

• None of these

$\int {\mathrm{e}}^{\mathrm{x}}\left(\frac{1}{\mathrm{x}} - \frac{1}{{\mathrm{x}}^2}\right)\mathrm{dx}$ is equal to

• $\frac{- {\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^2} + \mathrm{c}$

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

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{c}$

• $\frac{- {\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{c}$