Mathematics - Indefinite Integration Question with Solution | TestHub

The integral can be rewritten as $\int \frac{dx}{{\sin }^{3/2}x{\cos }^{5/2}x}=\int \frac{{\sec }^{4}x}{{\tan }^{3/2}x}dx$. Let $t=\tan x$. Then $dt={\sec }^{2}xdx$. We also know that ${\sec }^{2}x=1+{\tan }^{2}x=1+t^2$. Substituting these into the integral, we get:$$\int \frac{(1+t^2)}{t^{3/2}}dt=\int (t^{-3/2}+t^{1/2})dt$$ Integrating term by term:$$=\frac{t^{-3/2+1}}{-3/2+1}+\frac{t^{1/2+1}}{1/2+1}+C=\frac{t^{-1/2}}{-1/2}+\frac{t^{3/2}}{3/2}+C$$

$$=-\frac{2}{\sqrt{t}}+\frac{2}{3}{t}^{3/2}+C$$ Replacing $t$ with $\tan x$: $$=-2{\tan }^{-1/2}x+\frac{2}{3}{\tan }^{3/2}x+C$$