The solution of the differential equation xdy - ydx = $\left(\sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right)\mathrm{dx}$ is

• $\mathrm{y} - \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{Cx}}^2$

• $\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{Cx}}^2$

• $\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} + {\mathrm{Cx}}^{2 }= 0$

• None of the above

If ${\int }_0^{{\mathrm{t}}^2}\mathrm{x} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \frac{2}{5}{\mathrm{t}}^5,\mathrm{t} > 0, \mathrm{then} \mathrm{f}\left(\frac{4}{25}\right)$ is

• $\frac{2}{5}$

• $\frac{5}{2}$

• $- \frac{2}{5}$

• None of these

The value of interal I = $\int \frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\sqrt{1 + \sin \left(2\mathrm{x}\right)}}\mathrm{dx}$ is

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

• $\sqrt{\mathrm{x}}$

• x

• $\sqrt{1 + 2\mathrm{x}}$

The value of interal ${\int }_{- \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^3\left(\mathrm{x}\right){\cos }^3\left(\mathrm{x}\right)\mathrm{dx}$ is

• 0

• 1/2

• 1

• None of these

$\int \frac{{\mathrm{x}}^{\mathrm{e} - 1} + {\mathrm{e}}^{\mathrm{x} - 1}}{{\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}$ is equal to

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

• $\frac{1}{\mathrm{e}}\log \left({\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{c}$

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

• None of the above

426.

${\int }_0^1\frac{\mathrm{dx}}{\sqrt{1 + \mathrm{x}} + \sqrt{\mathrm{x}}}$ is equal to

• $\frac{4}{3}\left(\sqrt{2} - 1\right)$

• $\frac{3}{4}\left(\sqrt{2} - 1\right)$

• $\frac{4}{3}\left(1 - \sqrt{2}\right)$

• $\frac{3}{4}\left(1 - \sqrt{2}\right)$

${\int }_0^1\left|5\mathrm{x} - 3\right|\mathrm{dx}$ is equal to

• $\frac{10}{13}$

• $\frac{31}{10}$

• $\frac{13}{10}$

• None of these

${\int }_0^{\mathrm{\pi }}{\mathrm{xsin}}^4\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{3\mathrm{\pi }}{16}$

• $\frac{3{\mathrm{\pi }}^2}{16}$

• $\frac{16\mathrm{\pi }}{3}$

• $\frac{16{\mathrm{\pi }}^2}{3}$

$\int 5^{5^{5^{\mathrm{x}}}} 5^{5^{\mathrm{x}}} 5^{\mathrm{x}} \mathrm{dx}$ is equal to

• $5^{5^{5^{\mathrm{x}}}}{\left(\log \left(5\right)\right)}^3 + \mathrm{c}$

• $\frac{5^{5^{5^{\mathrm{x}}}}}{{\left(\log \left(5\right)\right)}^3} + \mathrm{c}$

• $\frac{5^{{}^{5^{\mathrm{x}}}}}{{\left(\log \left(5\right)\right)}^3} + \mathrm{c}$

• $5^{{}^{5^{\mathrm{x}}}}{\left(\log \left(5\right)\right)}^3 + \mathrm{c}$

The value of ${\int }_2^4\left\{\left|\mathrm{x} - 2\right| + \left|\mathrm{x} - 3\right|\right\}\mathrm{dx}$ is

• 1 sq unit

• 2 sq unit

• 3 sq unit

• 5 sq unit