Evaluating the Definite Integral $\int_0^{\pi}\cos^{2n} \theta d\theta$ $$\int_0^{\pi}\cos^{2n} \theta d\theta$$

$$u=\cos \theta \implies du= -\sin \theta d\theta \implies d\theta= -\frac{du}{1-u^2} $$

$$\int_{-1}^1 \frac{u^n}{1-u^2} du=\int_{-1}^1 \frac{u^n}{(1-u)(1+u)}du$$

I have no idea what to do next, any guidance is appreciated!
HINT

$\int_0^{\pi}\cos^{2n} \theta d\theta=2\int_0^{\pi/2}\cos^{2n} \theta d\theta$

which is Wallis' integral: integral
See this document (page 5 theorem 2.2): http://www.les-mathematiques.net/phorum/file.php?4,file=57836,filename=507-1088-1-PB.pdf