$\int {\cos }^{\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right){\sin }^{\raisebox{1ex}{$-11$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to:

• $\log \left|{\sin }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\right| + \mathrm{c}$

• $\frac{4}{7}{\tan }^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right) + \mathrm{c}$

• $\frac{- 7}{4}{\tan }^{\raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right) + \mathrm{c}$

• $\log \left|{\cos }^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.}\left(\mathrm{x}\right)\right| + \mathrm{c}$

$\int \frac{\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right)}{\sqrt{\sin \left(2\mathrm{\theta }\right)}}\mathrm{d\theta }$ is equal to :

• $\log \left|\cos \left(\mathrm{\theta }\right) - \sin \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right|$

• $\log \left|\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right) + \sqrt{\sin \left(2\mathrm{\theta }\right)}\right|$

• ${\sin }^{-1}\left(\sin \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right)\right)$

• ${\sin }^{-1}\left(\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)\right)$

${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{\mathrm{dx}}{1 + \sqrt{\tan \left(\mathrm{x}\right)}}$ is equal to :

• $\frac{\mathrm{\pi }}{12}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

• $2\mathrm{\pi }$

${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}\frac{{\sin }^4\left(\mathrm{x}\right)}{{\sin }^4\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)}$dx is equal to :

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

• $2\mathrm{\pi }$

The value of $\frac{2^{\sin \left(\mathrm{x}\right)}}{2^{\sin \left(\mathrm{x}\right)} + 2^{\cos \left(\mathrm{x}\right)}}\mathrm{dx}$ is :

• 2

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

• $2\mathrm{\pi }$

If f is continuous function, then :

• ${\int }_{- 2}^2\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_0^2\left[\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(- \mathrm{x}\right)\right]\mathrm{dx}$

• ${\int }_{- 3}^{5}2\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = {\int }_{- 6}^{10}\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

• ${\int }_{- 3}^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 4}^4\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

• ${\int }_{- 3}^5\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = {\int }_{- 2}^6\mathrm{f}\left(\mathrm{x} - 1\right)\mathrm{dx}$

67.

The area of the region bounded by y² = 4ax and x² = 4ay, a > 0 in sq unit, is :

• 16$\frac{{\mathrm{a}}^2}{3}$

• $14\frac{{\mathrm{a}}^2}{3}$

• $13\frac{{\mathrm{a}}^2}{3}$

• 16a²

An integrating factor of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{ylog}\left(\mathrm{x}\right) = {\mathrm{xe}}^{\mathrm{x}}{\mathrm{x}}^{\frac{1}{2}\log \left(\mathrm{x}\right)}$, (x > 0) is :

• x^log(x)

• ${\left(\sqrt{\mathrm{x}}\right)}^{\log \left(\mathrm{x}\right)}$

• ${\left(\sqrt{\mathrm{e}}\right)}^{{\left(\log \left(\mathrm{x}\right)\right)}^2}$

• ${\mathrm{e}}^{{\mathrm{x}}^2}$

The solution of ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{dy}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dx}$}\right.} = \left(\mathrm{x} + 1\right)$, y(0) = 3 is :

• y = xlog(x) - x + 2

• y = (x + 1)$\log \left|\mathrm{x} + 1\right| - \mathrm{x} + 3$

• $\mathrm{y} = \left(\mathrm{x} + 1\right)\log \left|\mathrm{x} + 1\right| + \mathrm{x} + 3$

• y = $\mathrm{xlog}\left(\mathrm{x}\right) + \mathrm{x} + 3$

Solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}\tan \left(\mathrm{y}\right) = \sin \left(\mathrm{x} + \mathrm{y}\right) + \sin \left(\mathrm{x} - \mathrm{y}\right)$ is :

• $\sec \left(\mathrm{y}\right) + 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

• $\sec \left(\mathrm{y}\right) - 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

• $\cos \left(\mathrm{y}\right) - 2\sin \left(\mathrm{x}\right) = \mathrm{c}$

• $\tan \left(\mathrm{y}\right) - 2\sec \left(\mathrm{y}\right) = \mathrm{c}$