$\int \left(\frac{4{\mathrm{e}}^{\mathrm{x}} - 25}{2{\mathrm{e}}^{\mathrm{x}} - 5}\right)\mathrm{dx} = \mathrm{Ax} + \mathrm{Blog}\left(2{\mathrm{e}}^{\mathrm{x}} - 5\right) + \mathrm{c}$, then

• A = 5 and B = 3

• A = 5 and B = - 3

• A = - 5 and B = 3

• A = - 5 and B = - 3

${\int }_{- \frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\log \left(\frac{2 - \sin \left(\mathrm{x}\right)}{2 + \sin \left(\mathrm{x}\right)}\right)\mathrm{dx}$ is equal to

• 1

• 3

• 2

• 0

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

• $\log \left(\mathrm{a}\right) . {\mathrm{a}}^{\mathrm{x} + {\tan }^{-1}\left(\mathrm{x}\right)} + \mathrm{c}$

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

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

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

If $\int \frac{\mathrm{f}\left(\mathrm{x}\right)}{\log \left(\sin \left(\mathrm{x}\right)\right)}\mathrm{dx} = \log \left[\log \left(\sin \left(\mathrm{x}\right)\right)\right] + \mathrm{c}$, then f(x) is equal to

• cot(x)

• tan(x)

• sec(x)

• csc(x)

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left(\frac{\sqrt[\mathrm{n}]{\sec \left(\mathrm{x}\right)}}{\sqrt[\mathrm{n}]{\sec \left(\mathrm{x}\right)} +\sqrt[\mathrm{n}]{\csc \left(\mathrm{x}\right)}}\right)\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

${\int }_0^1{\mathrm{xtan}}^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ =

• $\frac{\mathrm{\pi }}{4} + \frac{1}{2}$

• $\frac{\mathrm{\pi }}{4} - \frac{1}{2}$

• $\frac{1}{2} - \frac{\mathrm{\pi }}{4}$

• $- \frac{\mathrm{\pi }}{4} - \frac{1}{2}$

397.

If $\int \frac{1}{\sqrt{9 - 16{\mathrm{x}}^2}}\mathrm{dx} = {\mathrm{\alpha sin}}^{-1}\left(\mathrm{\beta x}\right) + \mathrm{c}$, then $\mathrm{\alpha } + \frac{1}{\mathrm{\beta }}$ =

• 1

• $\frac{7}{12}$

• $\frac{19}{12}$

• $\frac{9}{12}$

If ${\int }_0^{\frac{\mathrm{\pi }}{2}}\log \left(\cos \left(\mathrm{x}\right)\right)\mathrm{dx} = \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$, then ${\int }_0^{\frac{\mathrm{\pi }}{2}}\log \left(\sec \left(\mathrm{x}\right)\right)\mathrm{dx}$ =

• $\frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

• $1 - \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

• $1 + \frac{\mathrm{\pi }}{2}\log \left(\frac{1}{2}\right)$

• $\frac{\mathrm{\pi }}{2}\log \left(2\right)$

$\int \frac{1}{\left({\mathrm{x}}^2 + 4\right)\left({\mathrm{x}}^2 + 9\right)}\mathrm{dx} = {\mathrm{Atan}}^{-1}\left(\frac{\mathrm{x}}{2}\right) + {\mathrm{Btan}}^{-1}\left(\frac{\mathrm{x}}{3}\right) + \mathrm{C}$, then A - B =

• $\frac{1}{6}$

• $\frac{1}{30}$

• $- \frac{1}{30}$

• $- \frac{1}{6}$

If $\sqrt{\frac{\mathrm{x} - 5}{\mathrm{x} - 7}}\mathrm{dx} = \mathrm{A}\sqrt{{\mathrm{x}}^2 - 12\mathrm{x} +\hspace{0.17em}35}$ + $\log \left|\mathrm{x} - 6 + \sqrt{{\mathrm{x}}^2 - 12\mathrm{x} + 35}\right| + \mathrm{C}$, then A =

• - 1

• $\frac{1}{2}$

• $- \frac{1}{2}$

• 1