41.

$\int \frac{\sec \left(\mathrm{x}\right)\csc \left(\mathrm{x}\right)}{2\cot \left(\mathrm{x}\right) - \sec \left(\mathrm{x}\right)\csc \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\log \left|\sec \left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right)\right|$ + c

• $\log \left|\sec \left(\mathrm{x}\right) + \csc \left(\mathrm{x}\right)\right|$ + c

• $\frac{1}{2}\log \left|\sec \left(2\mathrm{x}\right) + \tan \left(2\mathrm{x}\right)\right|$ + c

• $\log \left|\sec \left(2\mathrm{x}\right) + \csc \left(2\mathrm{x}\right)\right|$ + c

If ${\int }_0^{\mathrm{a}}\left(2\mathrm{a} - \mathrm{x}\right)\mathrm{dx} = \mathrm{\mu } \mathrm{and} {\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{\lambda }$, then ${\int }_0^{2\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ equals

• $2\mathrm{\lambda } - \mathrm{\mu }$

• $\mathrm{\lambda } + \mathrm{\mu }$

• $\mathrm{\mu } - \mathrm{\lambda }$

• $\mathrm{\lambda } - 2\mathrm{\mu }$

$\int \tan \left({\sin }^{-1}\left(\mathrm{x}\right)\right)\mathrm{dx}$ is equal to

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}} + \mathrm{c}$

• $- \sqrt{1 - {\mathrm{x}}^2} + \mathrm{c}$

• $\frac{- \mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}} + \mathrm{c}$

• $\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}} + \mathrm{c}$

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

• $\frac{\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}{5} + \mathrm{c}$

• $\frac{{\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right)}^5}{5}$ + c

• $\frac{{\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right)}^4}{4}$ + c

• $\frac{{\left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right)}^5}{5}$ + c

$\int {\mathrm{e}}^{\sin \left(\mathrm{\theta }\right)}\left[\log \left(\sin \left(\mathrm{\theta }\right)\right) + {\csc }^2\left(\mathrm{\theta }\right)\right]\cos \left(\mathrm{\theta }\right)\mathrm{d\theta }$ is equal to

• $\int {\mathrm{e}}^{\sin \left(\mathrm{\theta }\right)}\left[\log \left(\sin \left(\mathrm{\theta }\right)\right) + {\csc }^2\left(\mathrm{\theta }\right)\right] + \mathrm{c}$

• ${\mathrm{e}}^{\sin \left(\mathrm{\theta }\right)}\left[\log \left(\sin \left(\mathrm{\theta }\right)\right) + \csc \left(\mathrm{\theta }\right)\right] + \mathrm{c}$

• ${\mathrm{e}}^{\sin \left(\mathrm{\theta }\right)}\left[\log \left(\sin \left(\mathrm{\theta }\right)\right) - \csc \left(\mathrm{\theta }\right)\right]$

• ${\mathrm{e}}^{\sin \left(\mathrm{\theta }\right)}\left[\log \left(\sin \left(\mathrm{\theta }\right)\right) - {\csc }^2\left(\mathrm{\theta }\right)\right]$

$\int {\mathrm{e}}^{3\log \left(\mathrm{x}\right)}{\left({\mathrm{x}}^4 + 1\right)}^{- 1}\mathrm{dx}$ is equal to

• e^3log(x) + c

• $\frac{1}{4}\log \left({\mathrm{x}}^4 + 1\right)$

• log(x⁴ + 1) + c

• $\frac{1}{2}\log \left({\mathrm{x}}^4 + 1\right) + \mathrm{c}$

If ${\int }_0^{\mathrm{\pi }}\mathrm{xf}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{dx} = \mathrm{A}{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{f}\left(\sin \left(\mathrm{x}\right)\right)\mathrm{dx}$, then A is

• 0

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

• 2$\mathrm{\pi }$

${\int }_{- 2}^2\left|\left[\mathrm{x}\right]\right|\mathrm{dx}$ is equal to

• 1

• 2

• 3

• 4

If $\int \frac{\sin \left(\mathrm{x}\right)}{\sin \left(\mathrm{x} - \mathrm{\alpha }\right)}\mathrm{dx} = \mathrm{Ax} + \mathrm{Blog}\left(\sin \left(\mathrm{x} - \mathrm{\alpha }\right)\right) + \mathrm{C}$, then value of A - B at $\mathrm{\alpha } = \frac{\mathrm{\pi }}{2}$ is

• - 1

• 1

• 2

• 0

The area bounded by $\mathrm{y} = {\sin }^{-1}\left(\mathrm{x}\right), \mathrm{x} = \frac{1}{\sqrt{2}}$and x - axis is

• $\left(\frac{1}{\sqrt{2}} + 1\right)\mathrm{sq} \mathrm{unit}$

• $\left(1 - \frac{1}{\sqrt{2}}\right)$ sq unit

• $\frac{\mathrm{\pi }}{4\sqrt{2}}$ sq unit

• $\left(\frac{\mathrm{\pi }}{4\sqrt{2}} + \frac{1}{\sqrt{2}} - 1\right) \mathrm{sq} \mathrm{unit}$