integrate $\tan^2x \sin x$

Question

How do I integrate (find the primitive to) $$ \int \tan^2 x \sin x dx$$

My approach has been to rewrite it to $$\int \frac{\sin^3 x}{\cos^2 x}dx$$ and the do the substitution $t = \tan x/2$. This works but I believe there's an easier way.

(I know that $0 < x < \pi/2$)

Answer 1

Let $\cos x = t$ , then the integral reduces to :

$$\int \frac{\sin^3 x}{\cos^2 x} \, dx = \int \frac{-(1-t^2)}{t^2} \, dt.$$

This can easily be solved.

Answer 2

HINT : $$ \int \tan^2 x \sin x\ dx=-\int (\sec^2 x-1)\ d(\cos x). $$