71.

If $\int {\mathrm{e}}^{\mathrm{x}}\left(\frac{1 - \sin \left(\mathrm{x}\right)}{1 - \cos \left(\mathrm{x}\right)}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) + \mathrm{constant}$, then f(x) is equal to

• ${\mathrm{e}}^{\mathrm{x}}\cot \left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• ${\mathrm{e}}^{-\mathrm{x}}\cot \left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $- {\mathrm{e}}^{\mathrm{x}}\cot \left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

• $- {\mathrm{e}}^{- \mathrm{x}}\cot \left(\frac{\mathrm{x}}{2}\right) + \mathrm{c}$

If I_n = $\int {\mathrm{x}}^{\mathrm{n}}{\mathrm{e}}^{\mathrm{cx}}\mathrm{dx}$ for n $\ge$ 1, then cI_n + n . I_{n - 1} is equal to

• xⁿe^cx

• xⁿ

• e^cx

• xⁿ + e^cx

If $\int {\mathrm{e}}^{\mathrm{x}}\left(1 + \mathrm{x}\right) . {\sec }^2\left({\mathrm{xe}}^{\mathrm{x}}\right)\mathrm{dx}$ = f(x) + constant, then f(x) is equal to

• $\cos \left({\mathrm{xe}}^{\mathrm{x}}\right)$

• $\sin \left({\mathrm{xe}}^{\mathrm{x}}\right)$

• $2{\tan }^{-1}\left(\mathrm{x}\right)$

• $\tan \left({\mathrm{xe}}^{\mathrm{x}}\right)$

${\int }_0^1{\mathrm{x}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{1 - \mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{9}$

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{16}$

${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sin \left|\mathrm{x}\right|\mathrm{dx}$ is equal to

• 0

• 1

• 2

• $\mathrm{\pi }$

The area (in sq unit) of the region bounded by the curves 2x = y² - 1 and x = 0 is

• $\frac{1}{3}$

• $\frac{2}{3}$

• 1

• 2

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{xy} + \mathrm{y}}{\mathrm{xy} + \mathrm{x}} \mathrm{is}$

• $\mathrm{x} + \mathrm{y} = \log \left(\frac{\mathrm{cy}}{\mathrm{x}}\right)$

• $\mathrm{x} + \mathrm{y} = \log \left(\mathrm{cxy}\right)$

• $\mathrm{x} - \mathrm{y} - \log \left(\frac{\mathrm{cx}}{\mathrm{y}}\right)$

• $\mathrm{y} - \mathrm{x} = \log \left(\frac{\mathrm{cx}}{\mathrm{y}}\right)$

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} - 2\mathrm{y} + 1}{2\mathrm{x} - 4\mathrm{y}} \mathrm{is}$

• (x - 2y)² + 2x = c

• (x - 2y)² + x = c

• (x - 2y)² + 2x² = c

• (x - 2y) + x² = c

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{ytan}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}}\sec \left(\mathrm{x}\right) \mathrm{is}$

• $\mathrm{y} = {\mathrm{e}}^{\mathrm{x}}\cos \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{ycos}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}} + \mathrm{c}$

• $\mathrm{y} = {\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{ysin}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}} + \mathrm{c}$

The solution of the differential equation

${\mathrm{xy}}^2\mathrm{dy} - \left({\mathrm{x}}^3 + {\mathrm{y}}^3\right)\mathrm{dx} = 0 \mathrm{is}$

• ${\mathrm{y}}^3 = 3{\mathrm{x}}^3 + \mathrm{c}$

• ${\mathrm{y}}^3 = 3{\mathrm{x}}^3 \log \left(\mathrm{cx}\right)$

• ${\mathrm{y}}^3 = 3{\mathrm{x}}^3 + \log \left(\mathrm{cx}\right)$

• ${\mathrm{y}}^3 +\hspace{0.17em}3{\mathrm{x}}^3 = \log \left(\mathrm{cx}\right)$