Question

integrar sen^4/cos^2​

Answer 1

Observa que:

$\sin^4 x=(\sin^2x)^2=(1-\cos^2x)^2 =1-2\cos^2 x+\cos^4 x$

Además, recordar la identidad $\cos^2 x=\frac{1+\cos 2x}{2}$.

Entonces:

$\int\frac{\sin^4x}{\cos^2 x}dx=\int\frac{1-2\cos^2 x+\cos^4 x}{\cos^2 x}dx\\\\=\int\frac{1}{\cos^2 x}dx-2\int dx+\int\cos^2 x\,dx\\\\=\int\sec^2 x\,dx-2\int dx+\int\frac{1+\cos 2x}{2}dx\\\\=\int\sec^2x\,dx-2\int dx+\frac{1}{2}\int dx+\frac{1}{2}\int\cos 2x\,dx\\\\=\tan x-2x+\frac{1}{2}x+\frac{1}{4}\sin 2x+C\\\\=\tan x-\frac{3}{2}x+\frac{1}{4}\sin 2x+C$