[Math] The integral $\int \frac{\sin^3(x)}{\cos^4(x)}\,dx$

Question

Hey I'm having problem solving this integral :

$$\int \frac{\sin^3(x)}{\cos^4(x)}dx$$

I think we should use $t=\sin(x)$ but it's not working for me.
and if I use $t=\tan(x/2)$ it gets worse.
Any ideas ?

Answer 1

Let $t=\cos x$ then $dt=-\sin x \, dx$ hence $$\int\frac{\sin^3x}{\cos^4x}dx=-\int\frac{1-t^2}{t^4}dt$$ can you take it from here?