51.

If I_n = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$, then $\frac{1}{{\mathrm{I}}_3 + {\mathrm{I}}_5}$ is

• 1/4

• 1/2

• 1/8

• 4

If ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^6\left(\mathrm{x}\right)\mathrm{dx} = \frac{5\mathrm{\pi }}{32}$, then the value of ${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}\left({\sin }^6\left(\mathrm{x}\right) + {\cos }^6\left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• $\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$

• $\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$16$}\right.$

• $\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{y}}{\mathrm{x}} + \frac{\mathrm{\varphi }\left({\displaystyle \frac{\mathrm{y}}{\mathrm{x}}}\right)}{\mathrm{\varphi }\text{'}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)}$ is

• $\mathrm{x\varphi }\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{k}$

• $\mathrm{\varphi }\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{kx}$

• $\mathrm{y\varphi }\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{k}$

• $\mathrm{\varphi }\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{ky}$

If the integrating factor of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{P}\left(\mathrm{x}\right)\mathrm{y} = \mathrm{Q}\left(\mathrm{x}\right)$ is x, then P(x) is

• x

• x²/2

• 1/x

• 1/x²

If c₁, c₂, c₃, c₄, c₅ and c₆  are constants, then the order of the differential equation whose general solution is given by y = c₁ cos(x + c₂) + c₃ sin(x + c₄) + c₅e^x + c₆, is

• 6

• 5

• 4

• 3

y = 2e^2x - e^{- x} is solution of the differential equation

• y₂ + y₁ + 2y = 0

• y₂ - y₁ + 2y = 0

• y₂ + y₁ = 0

• y₂ - y₁ - 2y = 0