How to do this integral $\int_{-\pi}^{\pi} x^n \cos^m(x) dx$? is there a way to explicitely evaluate this integral for natural numbers $n,m$:
$$\int_{-\pi}^{\pi} x^n \cos^m(x) dx.$$ 
Apparently, if $n$ is odd, this integral is zero due to symmetry.

$$
\int_{-\pi}^\pi x^{10}\cos^{10}(x)\;dx =
{\frac {49408448066608271851}{16986931200000000}}\,\pi -{\frac {
13747940134011979}{7077888000000}}\,{\pi }^{3}
+{\frac {
3845425458091}{9830400000}}\,{\pi }^{5}
-{\frac {157029277}{
4096000}}\,{\pi }^{7}
+{\frac {49133}{20480}}\,{\pi }
^{9}
+{\frac {63}{1408}}\,{\pi }^{11}
$$
Method: repeated integration by parts...
