21.

$\int \left(\sqrt{\tan \left(\mathrm{x}\right)} + \sqrt{\cot \left(\mathrm{x}\right)}\right)\mathrm{dx}$

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

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

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

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

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

• $\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• $\frac{\mathrm{x}\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• $\frac{\sin \left(\mathrm{x}\right) - \mathrm{xcos}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• None of these

${\int }_0^{\mathrm{x}}\frac{\mathrm{xdx}}{1 + \cos \left(\mathrm{\alpha }\right)\sin \left(\mathrm{x}\right)}, \left(0 < \mathrm{\alpha } < \mathrm{\pi }\right)$ is equal to

• $\frac{\mathrm{\pi \alpha }}{\sin \left(\mathrm{\alpha }\right)}$

• $\frac{\mathrm{\pi \alpha }}{\cos \left(\mathrm{\alpha }\right)}$

• $\frac{\mathrm{\pi \alpha }}{1 +\hspace{0.17em}\sin \left(\mathrm{\alpha }\right)}$

• $\frac{\mathrm{\pi \alpha }}{1 - \cos \left(\mathrm{\alpha }\right)}$

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\cos \left(\mathrm{x}\right)}{1 + {\mathrm{e}}^{\mathrm{x}}}d\mathrm{x}$

• 1

• 0

• - 1

• None of these

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

• $\mathrm{\pi }$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

The area bounded by the parabola y² = x, straight line y = 4 and y-axis is square units is

• 16/3 sq. unit

• 64/3 sq. unit

• $7\sqrt{2}$ sq. unit

• None of these

The differential equation of all circles which pass through the origin and whose centres lie on y-axis is

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

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

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

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

If m and n are order and degree of the equation

${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^5 + 4 . \frac{{\left({\displaystyle \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}}\right)}^3}{\left({\displaystyle \frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}}\right)} + \left(\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}\right) = {\mathrm{x}}^2 - 1$, then

• m = 3, n = 3

• m = 3, n = 2

• m = 3, n = 5

• m = 3, n = 1

The integrating factor of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{xlog}\left(\mathrm{x}\right)\right) + \mathrm{y} = 2\log \left(\mathrm{x}\right)$ is iven by

• e^x

• log(x)

• log(log(x))

• x

The differential equation whose solution is (x - h)² + (y - k)² = a² (a is a constant), is

• ${\left[1+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^3 = {\mathrm{a}}^2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$

• ${\left[1+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^3 = {\mathrm{a}}^2{\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^2$

• ${\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\right]}^3 = {\mathrm{a}}^2{\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^2$

• None of these