The value of $\int \frac{1}{1 + \cos \left(8\mathrm{x}\right)}\mathrm{dx}$ is

• $\frac{\tan \left(2\mathrm{x}\right)}{8} + \mathrm{c}$

• $\frac{\tan \left(8\mathrm{x}\right)}{8} + \mathrm{c}$

• $\frac{\tan \left(4\mathrm{x}\right)}{4} + \mathrm{c}$

• $\frac{\tan \left(4\mathrm{x}\right)}{8} + \mathrm{c}$

The value of $\int {\mathrm{e}}^{\mathrm{x}}\left({\mathrm{x}}^5 + 5{\mathrm{x}}^4 + 1\right)\mathrm{dx}$ is

• e^x . x⁵ + c

• e^x . x⁵ + e^x + c

• e^{x + 1} . x⁵ + c

• 5x⁴ . e^x + c

The value of $\int \frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2 - 1}\mathrm{dx}$ is

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

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

• $\mathrm{x} +\hspace{0.17em}\log \left(\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right) + \mathrm{c}$

• 5x⁴ . e^x + c

The area bounded by the curve x = 4 - y² and the Y-axis is

• 16 sq units

• 32 sq units

• $\frac{32}{3} \mathrm{sq} \mathrm{units}$

• $\frac{16}{3} \mathrm{sq} \mathrm{units}$

25.

The differential equation of the family of straight lines whose slope is equal to y-intercept is

• $\left(\mathrm{x} + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = 0$

• $\left(\mathrm{x} + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} - 1}{\mathrm{y} - 1}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} + 1}{\mathrm{y} + 1}$

The order and degree of the differential equation ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^5\right]}^{\frac{1}{3}} = \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ are respectively

• 1, 5

• 2, 1

• 2, 5

• 2, 3