Often with integration by substitution I see (and use) the notation $ x \to \frac{\pi}{2} - x $, for the simple reason that I don't have to rename the variable that I am integrating with respect to, but recently this notation has gotten me confused, in the sense that I'm not sure I properly understand it.
For example, with the integral $$ \int^{2\pi}_{0} \frac{x\sin x}{3+\sin^2 x} \ dx $$
I proceeded as follows.
First I will note, that previously I have shown that $\int^{\pi}_{0} \frac{x \sin x}{3+\sin^2 x} \ dx =\frac{\pi}{2}\int^{\pi}_{0} \frac{\sin x}{3+\sin^2 x} \ dx = \frac{\pi}{4}\ln 3 $
Returning to the integral $$ \int^{2\pi}_{0} \frac{x\sin x}{3+\sin^2 x} \ dx = \int^{\pi}_{0} \frac{x\sin x}{3+\sin^2 x} \ dx + \int^{2\pi}_{\pi} \frac{x\sin x}{3+\sin^2 x} \ dx $$
Now in the second integral substitute $ x \to x - \pi $ then we have
$$\int^{2\pi}_{\pi} \frac{x\sin x}{3+\sin^2 x} \ dx = \int^{\pi}_{0} \frac{(x - \pi)(-\sin x)}{3+\sin^2 x} \ dx= -\int^{\pi}_{0} \frac{x\sin x}{3+\sin^2 x}\ dx + \int^{\pi}_{0} \frac{\pi \sin x}{3+\sin^2 x}\ dx $$
This means that
$$ \int^{2\pi}_{0} \frac{x\sin x}{3+\sin^2 x} \ dx = \int^{\pi}_{0} \frac{\pi \sin x}{3+\sin^2 x}\ dx = \frac{\pi}{2} \ln 3 $$ by the initial result.
This is incorrect; the answer should be $ - \frac{\pi}{2} \ln 3 $, but I don't see my error. I think an error may have arisen with my substitution, so I'd appreciate it if someone could point out where the error is and why it is wrong.
Also, I wasn't sure how to entitle this question, but I hope that the title I've chosen is appropriate.
When you make the substitution $ x \to x - \pi $ then you rather have
$$\int^{2\pi}_{\pi} \frac{x\sin x}{3+\sin^2 x} \ dx = \int^{\pi}_{0} \frac{\color{red}{(x+ \pi)}(-\sin x)}{3+\sin^2 x} \ dx= -\int^{\pi}_{0} \frac{x\sin x}{3+\sin^2 x}\ dx \color{red}{- }\int^{\pi}_{0} \frac{\pi \sin x}{3+\sin^2 x}\ dx $$ which gives the correct answer.