Dividing the interval (1, 2) into four equal parts and using Simpson's rule, the value of ${\int }_1^2\frac{\mathrm{dx}}{\mathrm{x}}$ will be

• 0.6932

• 0.6753

• 0.6692

• 0.7132

Taking four subintervals, the value of ${\int }_0^1\frac{\mathrm{dx}}{1 + \mathrm{x}}$ by usin trapezoidal rule will be

• 0.6870

• 0.6677

• 0.6970

• 0.5970

$\int \left(\frac{2018{\mathrm{x}}^{2017} + {2018}^{\mathrm{x}} {\log }_{\mathrm{e}}\left(2018\right)}{{\mathrm{x}}^{2018} + {2018}^{\mathrm{x}}}\right)\mathrm{dx}$ =

• $\log \left|{2018}^{\mathrm{x}} + {\mathrm{x}}^{2018}\right| + \mathrm{c}$

• ${\left({2018}^{\mathrm{x}} + {\mathrm{x}}^{2018}\right)}^{-1} + \mathrm{c}$

• $\left|{2018}^{\mathrm{x}} + {\mathrm{x}}^{2018}\right| + \mathrm{c}$

• ${2018}^{\mathrm{x}} + {\mathrm{x}}^{2018} + \mathrm{c}$

$\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 + 4\mathrm{x} +\hspace{0.17em}5}$ =

• ${\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{c}$

• ${\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{c}\left(\mathrm{x} + 2\right)$

• ${\tan }^{-1}\left(\mathrm{x} + 2\right) + \mathrm{c}$

• $\left(\mathrm{x} + 2\right){\tan }^{-1}\left(\mathrm{x} + 2\right) + \mathrm{c}$

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left({\cos }^2\left(\frac{\mathrm{x}}{2}\right) - {\sin }^2\left(\frac{\mathrm{x}}{2}\right)\right)$ =

• 0

• 1

• - 1

• None of the above

Evaluate ${\int }_{- 2}^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$, where f(x) = $$\begin{gathered} \left\{1 - 2\mathrm{x}, \mathrm{x} \le 0 \\ 1 + 2\mathrm{x}, \mathrm{x} \ge 0\right\} \end{gathered}$$

• 0

• 2

• 4

• 6

$\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x}}\left(\mathrm{x} +\hspace{0.17em}9\right)}$ is equal to

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

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

• ${\tan }^{-1}\left(\sqrt{\mathrm{x}}\right) + \mathrm{C}$

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

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

• xe^x + C

• x²x^x + C

• (x + 1)e^x + C

• (x² + 1)e^x + C

619.

$\int \frac{\mathrm{dx}}{{\mathrm{a}}^2{\sin }^2\left(\mathrm{x}\right) + {\mathrm{b}}^2{\cos }^2\left(\mathrm{x}\right)}$ is equal to

• $\frac{1}{\mathrm{ab}}{\tan }^{-1}\left(\frac{\mathrm{atan}\left(\mathrm{x}\right)}{\mathrm{b}}\right) + \mathrm{C}$

• ${\tan }^{-1}\left(\frac{\mathrm{atan}\left(\mathrm{x}\right)}{\mathrm{b}}\right) + \mathrm{C}$

• $\frac{1}{\mathrm{ab}}{\tan }^{-1}\left(\frac{\mathrm{btan}\left(\mathrm{x}\right)}{\mathrm{a}}\right) + \mathrm{C}$

• ${\tan }^{-1}\left(\frac{\mathrm{btan}\left(\mathrm{x}\right)}{\mathrm{a}}\right) + \mathrm{C}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^8\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{512}$

• $\frac{3\mathrm{\pi }}{512}$

• $\frac{5\mathrm{\pi }}{512}$

• $\frac{7\mathrm{\pi }}{512}$