51.

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

• $\tan \left(\mathrm{x}\right) + \mathrm{C}$

• $\sec \left(\mathrm{x}\right) + \mathrm{C}$

• $2\mathrm{x} - \sec \left(\mathrm{x}\right) + \mathrm{C}$

• 2x - $\tan \left(\mathrm{x}\right)$ + C

The value of ${\int }_0^6\left|\mathrm{x} - 3\right|d\mathrm{x}$ is equal to

• 6

• 0

• 12

• 9

If f(x) = ${\int }_{2\mathrm{x}}^{\sin \left(\mathrm{x}\right)}\cos \left({\mathrm{t}}^3\right)\mathrm{dt}$, then f'(x) is equal to

• $\cos \left({\sin }^3\left(\mathrm{x}\right)\right)\cos \left(\mathrm{x}\right) - 2\cos \left(8{\mathrm{x}}^3\right)$

• $\sin \left({\sin }^3\left(\mathrm{x}\right)\right)\sin \left(\mathrm{x}\right) - 2\sin \left(8{\mathrm{x}}^3\right)$

• $\cos \left({\cos }^3\left(\mathrm{x}\right)\right)\cos \left(\mathrm{x}\right) - 2\cos \left({\mathrm{x}}^3\right)$

• $\cos \left({\sin }^3\left(\mathrm{x}\right)\right) - \cos \left(8{\mathrm{x}}^3\right)$

${\int }_0^{10}\frac{{\mathrm{x}}^{10}}{{\left(10 - \mathrm{x}\right)}^{10} + {\mathrm{x}}^{10}}\mathrm{dx}$ is equal to

• 10

• 5

• 2

• $\frac{1}{2}$

The area bounded by y = x + 2, y = 2 - x and the x-axis is (in square units)

• 1

• 2

• 4

• 6

The value of ${\int }_0^1\sqrt{\mathrm{x}}{\mathrm{e}}^{\sqrt{\mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{\left(\mathrm{e} - 2\right)}{2}$

• 2(e - 2)

• 2e - 1

• 2(e - 1)

The order and degree of the differential equation ${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^3}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = 2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \sqrt[3]{{\cos }^2\left(\mathrm{x}\right)}$ are, respectively

• 3 and 1

• 3 and 3

• 1 and 3

• 3 and 2

The differential equation representing the family of curves given by y = ae^- 3x + b, where a and b are arbitrary constants, is

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 3\frac{\mathrm{dy}}{\mathrm{dx}} - 2\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 3\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 3\frac{\mathrm{dy}}{\mathrm{dx}} - 2\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 3\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

An integrating factor of the differential equation xdy - ydx + x²e^xdx = 0 is

• $\frac{1}{\mathrm{x}}$

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

• $\sqrt{1 + {\mathrm{x}}^2}$

• x

The solution of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{y}}{1 + \log \left(\mathrm{x}\right)}$ is

• y = log(x) + C

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

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

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