Mathematics - Indefinite Integration Question with Solution | TestHub

$I=\int \frac{{\cot }^{4}x{\mathrm{cosec}}^{2}xdx}{{\left(1+{\cot }^{5}x\right)}^{3/5}}$

$1+{\cot }^5\mathrm{x}=\mathrm{t}$

$\mathrm{dt}=-5{\cot }^4\mathrm{x}\cdot {\mathrm{cosec}}^2\mathrm{x}\mathrm{dx}$

$I=-\frac{1}{5}\int \frac{\mathrm{dt}}{{\mathrm{t}}^{3/5}}$

$$\begin{array}{l}I=-\frac{1}{5}\frac{5t^{2/5}}{2}+C\Rightarrow I=-\frac{1}{2}{\left(1+{\cot }^{5}x\right)}^{2/5}+C\end{array}$$

$\left.\begin{array}{r}A=5\\ B=2/5\end{array}\right\}\Rightarrow A·B=2$