If $\int \sqrt{\frac{x - 5}{x - 7}} dx = A \sqrt{x^2 - 12 x + 35} + \log |x - 6 + \sqrt{x^2 - 12 x + 35}| + C$,then $A = . . . . . .$

$\int \operatorname{cosec}^5 x \, dx =$

Find $\int [\sqrt{\cot x} + \sqrt{\tan x}] \, dx$.

If $\int \frac{dx}{(x^2+9) \sqrt{x^2+16}} = \frac{1}{3 \sqrt{7}} \operatorname{Tan}^{-1} \left( K \frac{x}{\sqrt{16+x^2}} \right) + c$, then $K=$

If $I_n = \int \tan^n x \, dx$ $(n > 1)$,then $I_4 + I_6 =$

For real numbers $\alpha, \beta, \gamma$ and $\delta,$ if $\int \frac{\left(x^{2}-1\right)+\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)}{\left(x^{4}+3 x^{2}+1\right) \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)} d x =\alpha \log _{e}\left(\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)\right) +\beta \tan ^{-1}\left(\frac{\gamma\left(x^{2}-1\right)}{x}\right)+\delta \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)+C$ where $C$ is an arbitrary constant,then the value of $10(\alpha+\beta \gamma+\delta)$ is equal to ....... .