51.

$\int \frac{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)}{1 + 2\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

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

• $\frac{\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)} + \mathrm{c}$

• $- \frac{1}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}$

• $\frac{\mathrm{x}}{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{c}$

$\int \frac{1}{\mathrm{x}}{\log }_{\mathrm{ex}}\left(\mathrm{e}\right)\mathrm{dx}$ is equal to

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

• ${\log }_{\mathrm{e}}\left({\log }_{\mathrm{e}}\left(\mathrm{ex}\right) - 1\right)$ + c

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

• ${\log }_{\mathrm{e}}\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right) + \mathrm{c}$

The value of ${\int }_1^{\mathrm{e}}{10}^{{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}d\mathrm{x}$ is equal to

• 10${\log }_{\mathrm{e}}\left(10\mathrm{e}\right)$

• $\frac{10\mathrm{e} - 1}{{\log }_{\mathrm{e}}\left(10\mathrm{e}\right)}$

• $\frac{10\mathrm{e}}{{\log }_{\mathrm{e}}\left(10\mathrm{e}\right)}$

• $\left(10\mathrm{e}\right){\log }_{\mathrm{e}}\left(10\mathrm{e}\right)$

The area between the curve y = 1 - $\left|\mathrm{x}\right|$ and the x-axis is equal to

• 1 sq unit

• $\frac{1}{2}$ sq unit

• $\frac{1}{3}$

• 2 sq unit

The value of ${\int }_{{\mathrm{e}}^{-1}}^{\mathrm{e}}\frac{\mathrm{dt}}{\mathrm{t}\left(1 + \mathrm{t}\right)}$ is equal to

• 0

• $\log \left(\frac{\mathrm{e}}{1 + \mathrm{e}}\right)$

• $\log \left(\frac{1}{1 + \mathrm{e}}\right)$

• 1

The figure shows a triangle AOB and the parabola y = x². The ratio of the area of the triangle AOB to the area of the region AOB of the parabola y = x is equal to

• $\frac{3}{5}$

• $\frac{3}{4}$

• $\frac{7}{8}$

• $\frac{5}{6}$

The value of ${\int }_{- 2}^4\left|\mathrm{x} + 1\right|\mathrm{dx}$ is equal to

• 12

• 14

• 13

• 16

The solution of $\cos \left(\mathrm{y}\right)\frac{\mathrm{dy}}{\mathrm{dx}} = {\mathrm{e}}^{\mathrm{x} + \sin \left(\mathrm{y}\right)} + {\mathrm{x}}^2{\mathrm{e}}^{\sin \left(\mathrm{y}\right)}$ is

• ${\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{- \sin \left(\mathrm{y}\right)}+ \frac{{\mathrm{x}}^3}{3} = \mathrm{c}$

• ${\mathrm{e}}^{- \mathrm{x}} - {\mathrm{e}}^{- \sin \left(\mathrm{y}\right)}+ \frac{{\mathrm{x}}^3}{3} = \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \sin \left(\mathrm{y}\right)}+ \frac{{\mathrm{x}}^3}{3} = \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{\sin \left(\mathrm{y}\right)}+ \frac{{\mathrm{x}}^3}{3} = \mathrm{c}$

The order and degree of the differential equation

${\left(1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right)}^{\frac{3}{4}} = {\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^{\frac{1}{3}}$

• (2, 4)

• (2, 3)

• (6, 4)

• (6, 9)

The integrating factor of the differential equation $\left(\mathrm{ylog}\left(\mathrm{y}\right)\right)\mathrm{dx} = \left(\log \left(\mathrm{y}\right) - \mathrm{x}\right)\mathrm{dy}$ is

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

• $\log \left(\log \left(\mathrm{y}\right)\right)$

• $1 + \log \left(\mathrm{y}\right)$

• $\log \left(\mathrm{y}\right)$