$\int \frac{3 - {\mathrm{x}}^2}{1 - 2\mathrm{x} + {\mathrm{x}}^2}{\mathrm{e}}^{\mathrm{x}}\mathrm{dx} = {\mathrm{e}}^{\mathrm{x}}\mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)$

• $\frac{1 + \mathrm{x}}{1 - \mathrm{x}}$

• $\frac{1 - \mathrm{x}}{1 + \mathrm{x}}$

• $\frac{1 - \mathrm{x}}{\mathrm{x} - 1}$

• $\frac{\mathrm{x} - 1}{1 + \mathrm{x}}$

$\int \frac{\sqrt{\cot \left(\mathrm{x}\right)}}{\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}\mathrm{dx} = - \mathrm{f}\left(\mathrm{x}\right) + \mathrm{c} \Rightarrow \mathrm{f}\left(\mathrm{x}\right)$

• $2\sqrt{\tan \left(\mathrm{x}\right)}$

• $- 2\sqrt{\tan \left(\mathrm{x}\right)}$

• $- 2\sqrt{\cot \left(\mathrm{x}\right)}$

• $2\sqrt{\cot \left(\mathrm{x}\right)}$

${\int }_{\frac{- \mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\log \left(\frac{2 - \sin \left(\mathrm{\theta }\right)}{2 + \sin \left(\mathrm{\theta }\right)}\right)\mathrm{d\theta }$ is equal to

• 0

• 1

• 2

• - 1

The area bouned by y = x² + 2, x-axis, x = 1 and x = 2 is

• $\frac{16}{3}$

• $\frac{17}{3}$

• $\frac{13}{3}$

• $\frac{20}{3}$

${\int }_0^2\frac{2\mathrm{x} - 2}{2\mathrm{x} - {\mathrm{x}}^2}\mathrm{dx}$ is equal to

• 0

• 2

• 3

• 4

76.

Integrating factor of (x + 2y³)$\frac{\mathrm{dy}}{\mathrm{dx}}$ = y² is

• ${\mathrm{e}}^{\left(\frac{1}{\mathrm{y}}\right)}$

• ${\mathrm{e}}^{- \left(\frac{1}{\mathrm{y}}\right)}$

• y

• $\frac{- 1}{\mathrm{y}}$

y = Ae^x + Be^2x + Ce^3x satisfies the differential equation

• y''' - 6y'' + 11y' - 6y = 0

• y''' + 6y'' + 11y' + 6y = 0

• y''' + 6y'' - 11y' + 6y = 0

• y''' - 6y'' - 11y' + 6y = 0

Observe the following statements

A. Integrating factor of $\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} + \mathrm{y} = {\mathrm{x}}^2$ is e^x

R. Integrating factor of $\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} + \mathrm{P}\left(\mathrm{x}\right)\mathrm{y} = \mathrm{Q}\left(\mathrm{x}\right)$ is ${\mathrm{e}}^{\int \mathrm{P}\left(\mathrm{x}\right)\mathrm{dx}}$

• A is true, R is false

• A is false, R is true

• A is true, R is true, R $\Rightarrow$ A

• Both are false

For any integer n > 1, the number of positive divisors of n is denoted by d(n). Then, for a prime P, d (d (d(P)⁷)) is equal to

• 1

• 2

• 3

• p

$\sum _{\mathrm{k} = 1}^5\frac{1^3 + 2^3 + .... + {\mathrm{k}}^3}{1 + 3 + 5 + ... + \left(2\mathrm{k} - 1\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 22.5

• 24.5

• 28.5

• 32.5