Mathematics : Integrals

The value of ${\int }_0^5\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}$ by chosing six sub-intervals and by using Simpson's rule will be

• 1.3562

• 1.3662

• 1.3456

• 1.2662

If $\int \frac{\cos \left(4\mathrm{x}\right) + 1}{\cot \left(\mathrm{x}\right) - \tan \left(\mathrm{x}\right)}\mathrm{dx} = \mathrm{Acos}\left(4\mathrm{x}\right) + \mathrm{B}$, then the value of A is

• $\frac{1}{2}$

• $\frac{1}{8}$

• $- \frac{1}{8}$

• $\frac{1}{4}$

If f(x) = $\mathrm{f}\left(\mathrm{x}\right) = \sqrt{\mathrm{x}}, \mathrm{g}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}} - 1$ and $\int \mathrm{fog}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{Afog}\left(\mathrm{x}\right) + {\mathrm{Btan}}^{-1}\left(\mathrm{fog}\left[\mathrm{x}\right]\right) + \mathrm{C}$, then the value of A + B is

• 1

• 2

• 3

• None of these

The value of ${\int }_0^1{\tan }^{-1}\left(\frac{2\mathrm{x} - 1}{1 + \mathrm{x} - {\mathrm{x}}^2}\right)\mathrm{dx}$ is

• 0

• 1

• - 1

• None of these

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left(2\mathrm{x}\right){\tan }^{-1}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$ =

• 1

• 0

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2} - 1$

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