Integrating with universal substitution

Question

$$\int \frac{1-3\sin(2x)}{1+\cos(2x)}dx $$

I have a homework question solving this problem with universal substitution.

Question is - can I automatically put the substitutions even if in the $\sin$ and $\cos$ there is $2x$ inside instead of "classic" $x$?

for example if it was $3\sin(x)$ then $\sin x = 2u/(1+u^2)$?

Answer 1

Yes, you can just replace $\sin2x=\frac{2u}{1+u^2}$. To convince yourself, just do $z=2x$ first.

Answer 2

Using that $$\sin(2x)=2\sin(x)\cos(x)$$ and $$\cos(2x)=2\cos^2(x)-1$$ we get for the integral $$\int\frac{1}{2\cos^2(x)}-\frac{3\sin(x)}{\cos(x)}dx$$