51.

The integral ${\int }_0^1\frac{2{\sin }^{-1}\left({\displaystyle \frac{\mathrm{x}}{2}}\right)}{\mathrm{x}}\mathrm{dx}$ equals

• ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\frac{\mathrm{xdx}}{\tan \left(\mathrm{x}\right)}$

• ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\frac{2}{\tan \left(\mathrm{x}\right)}\mathrm{dx}$

• ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{2\mathrm{xdx}}{\tan \left(\mathrm{x}\right)}$

• ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\frac{2\mathrm{x}}{\sin \left(\mathrm{x}\right)}\mathrm{dx}$

The area of the plane region bounded by the curve x = y² - 2 and the· line y = - x is (in square units)

• $\frac{13}{3}$

• $\frac{2}{5}$

• $\frac{9}{2}$

• $\frac{5}{2}$

If ${\int }_0^{\mathrm{a}}\mathrm{f}\left(2\mathrm{a} - \mathrm{x}\right)\mathrm{dx} = \mathrm{m} \mathrm{and} {\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{n}$, then ${\int }_0^{2\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• 2m + n

• m + 2n

• m - n

• m + n

${\int }_{- 100}^{100}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• ${\int }_{- 100}^{100}\mathrm{f}\left({\mathrm{x}}^2\right)\mathrm{dx}$

• ${\int }_{- 100}^{100}\mathrm{f}\left(- {\mathrm{x}}^2\right)\mathrm{dx}$

• ${\int }_{- 100}^{100}\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)\mathrm{dx}$

• ${\int }_{- 100}^{100}\mathrm{f}\left(- \mathrm{x}\right)\mathrm{dx}$

${\int }_{- 1}^1\left({\mathrm{e}}^{{\mathrm{x}}^3} + {\mathrm{e}}^{- {\mathrm{x}}^3}\right)\left({\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{- \mathrm{x}}\right)\mathrm{dx}$ is equal to

• $\frac{{\mathrm{e}}^2}{2} - 2\mathrm{e}$

• e² - 2e

• 2(e² - e)

• 0

The family of curves y = e^asin(x), where a is anarbitrary constant, is represented by thedifferential equation 

• $\log \left(\mathrm{y}\right) = \tan \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}}$

• $\mathrm{ylog}\left(\mathrm{y}\right) = \tan \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}}$

• $\mathrm{ylog}\left(\mathrm{y}\right) = \sin \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}}$

• $\log \left(\mathrm{y}\right) = \cos \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}}$

The integrating factor of $\mathrm{x} \frac{d\mathrm{y}}{d\mathrm{x}} + \left(1 + \mathrm{x}\right) \mathrm{y} = \mathrm{x}$ is

• x

• 2x

• e^xlog(x)

• xe^x

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + 1 = {\mathrm{e}}^{\mathrm{x} + \mathrm{y}}$ is

• x + e^{x + y} = c

• x - e^{x + y} = c

• x + e^{- (x + y)} = c

• x - e^{- (x + y)} = c

The degree and order of the differential equation $\mathrm{y} = \mathrm{px} + \sqrt[3]{{\mathrm{a}}^2{\mathrm{p}}^2 + {\mathrm{b}}^2},$ where p = $\frac{\mathrm{dy}}{\mathrm{dx}}$, are respectively.

• 3, 1

• 1, 3

• 1, 1

• 3, 3

If the distance between (2, 3) and (- 5, 2) is equal to the distance between (x, 2) and (1, 3), then the values of x are

• - 6, 8

• 6, 8

• - 8, 6

• - 7, 7