$\int \frac{1 + \mathrm{x} + \sqrt{\mathrm{x} + {\mathrm{x}}^2}}{\sqrt{\mathrm{x}} + \sqrt{1 + \mathrm{x}}}\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{2}\sqrt{1 + \mathrm{x}} + \mathrm{C}$

• $\frac{2}{3}{\left(1 + \mathrm{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + \mathrm{C}$

• $\sqrt{1 + \mathrm{x}} + \mathrm{C}$

• $2{\left(1\hspace{0.17em}+ \mathrm{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} +\mathrm{C}$

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

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

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

• $- {\mathrm{xe}}^{\mathrm{x} + {\mathrm{x}}^{- 1}} + \mathrm{C}$

• ${\mathrm{xe}}^{\mathrm{x} + {\mathrm{x}}^{- 1}} + \mathrm{C}$

${\int }_{- 2}^2\left|\begin{array}{c}\left[\mathrm{x}\right]\end{array}\right|\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 1

• 2

• 3

• 4

${\int }_0^1\sin \left(2{\tan }^{-1}\left(\sqrt{\frac{1 + \mathrm{x}}{1 - \mathrm{x}}}\right)\right)\mathrm{dx} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

${\int }_0^3\frac{3\mathrm{x} + 1}{{\mathrm{x}}^2 + 9}\mathrm{dx}$ is equal to :

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{12}$

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{2}$

• $\log \left(2\sqrt{2}\right) + \frac{\mathrm{\pi }}{6}$

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

76.

If [2, 6] is divided into four intervals of equal length, then the approximate value of ${\int }_2^6\frac{1}{{\mathrm{x}}^2 - \mathrm{x}}\mathrm{dx}$ using Simpson's rule, is

• 0.3222

• 0.2333

• 0.5222

• 0.2555

The differential equation of the family of parabola with focus as the origin and the axis as X-axis, is

• $\mathrm{y}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + 4\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = 4\mathrm{y}$

• $- \mathrm{y}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 2\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y}$

• $\mathrm{y}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + \mathrm{y} = 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}$

• $\mathrm{y}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

Soution of $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}}{\sin \left(\mathrm{y}\right){\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle \mathrm{ycos}}{\displaystyle \left(\mathrm{y}\right)}}$ is

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2\log \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2 + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2 + \log \left(\mathrm{x}\right)$

• $\mathrm{ysin}\left(\mathrm{y}\right) = \mathrm{xlog}\left(\mathrm{x}\right) + \mathrm{C}$

The general solution of ${\mathrm{y}}^2\mathrm{dx} + \left({\mathrm{x}}^2 - \mathrm{xy} + {\mathrm{y}}^2\right)\mathrm{dy} = 0$ is :

• ${\tan }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \log \left(\mathrm{y}\right) + \mathrm{C}$

• $2{\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \log \left(\mathrm{x}\right) + \mathrm{C} = 0$

• $\log \left(\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right) + \log \left(\mathrm{y}\right) + \mathrm{C} = 0$

• ${\sinh }^{-1}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \log \left(\mathrm{y}\right) + \mathrm{C} = 0$

$\mathrm{The} \mathrm{solution} \mathrm{set} \mathrm{of} \left(5 + 4\cos \left(\mathrm{\theta }\right)\right)\left(2\cos \left(\mathrm{\theta }\right) + 1\right) = 0 \mathrm{in} \mathrm{the} \mathrm{interval} \left[0, 2\mathrm{\pi }\right], \mathrm{is} :$

• $\left\{\frac{\mathrm{\pi }}{3}, \frac{2\mathrm{\pi }}{3}\right\}$

• $\left\{\frac{\mathrm{\pi }}{3}, \mathrm{\pi }\right\}$

• $\left\{\frac{2\mathrm{\pi }}{3}, \frac{4\mathrm{\pi }}{3}\right\}$

• $\left\{\frac{2\mathrm{\pi }}{3}, \frac{5\mathrm{\pi }}{3}\right\}$