41.

Area bounded by the curves y = e^x, y = e^{- x} and the straight line x = 1 is (in sq units)

• $\mathrm{e} + \frac{1}{\mathrm{e}}$

• $\mathrm{e} + \frac{1}{\mathrm{e}} + 2$

• $\mathrm{e} + \frac{1}{\mathrm{e}} - 2$

• $\mathrm{e} - \frac{1}{\mathrm{e}} + 2$

The value of the integral ${\int }_1^{\mathrm{e}}\frac{1 + \log \left(\mathrm{x}\right)}{3\mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{3}{4}$

• e

The value of the integtral ${\int }_0^1\frac{{\mathrm{x}}^3}{1 + {\mathrm{x}}^8}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{6}$

The value of ${\int }_2^4\left(\frac{\log \left(\mathrm{t}\right)}{\mathrm{t}}\right)\mathrm{dt}$ is

• $\frac{1}{2}{\left(\log \left(2\right)\right)}^2$

• $\frac{5}{2}{\left(\log \left(2\right)\right)}^2$

• $\frac{3}{2}{\left(\log \left(2\right)\right)}^2$

• ${\left(\log \left(2\right)\right)}^2$

The solution of the differential equation (kx - y² ) dy = (x² - ky) dx is

• x³ - y³ = 3kxy + C

• x³ + y³ = 3kxy + C

• x² - y² = 2kxy + C

• x² + y² = 2kxy + C

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

• y = e^x + C

• y = x + e^x + C

• y = xe^x + C

• y = x(e^x + 1) + C

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

• 1, 1

• 1, 2

• 1, 3

• 2, 2

An integrating factor of the differential equation $\sin \left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{ycos}\left(\mathrm{x}\right) = 1$ is

• sin²(x)

• $\frac{2}{\sin \left(\mathrm{x}\right)}$

• $\log \left|\sin \left(\mathrm{x}\right)\right|$

• $\frac{1}{{\sin }^2\left(\mathrm{x}\right)}$