How to take the integral $\;\int\frac{\sin^2x\,dx}{\cos^4x}$

Question

There's

$$ \int \frac{\sin^2x}{\cos^4x} \,dx$$

Is there the way to solve it bypass general substitute like $\,t = \tan\dfrac x2$?

Answer 1

$$ \int \frac{\sin^2x \operatorname{dx}}{\cos^4x}= \int \frac{\tan^2x \operatorname{dx}}{\cos^2x}=\int \tan^2 x\sec^2x dx=\frac{\tan^3x}{3}+C $$

Answer 2

By parts:

$$\begin{cases}u=\sin x,&u'=\cos x\\{}\\v'=\frac{\sin x}{\cos^4x},&v=-\frac1{3\cos^3}\end{cases}\implies\int\frac{\sin^2x}{\cos^4x}dx=-\frac13\frac{\sin x}{\cos^2x}+\frac13\int\frac{dx}{\cos^2x}$$

and now remember that $\;(\tan x)'=\frac1{\cos^2x}\;$