Evaluating a specific definite integral

Question

Can someone help me evaluate the following integral?

$$\frac{1}π\int_{-π}^π x^{2n}cos(\frac{nπx}L) dx$$

Integration by parts does not seem to work. $$\frac{1}π\int_{-π}^π x^{2n}cos(\frac{nπx}L) dx = \frac{1}π([x^{2n}sin(\frac{nπx}L)\frac{L}{nπ}]_{-π}^π-\frac{2L}π\int_{-π}^πx^{2n-1}sin(\frac{nπx}L)dx)$$ If I continue to integrate by parts, then I won't arrive at an answer as I will keep getting powers of x.

What do you obtain with integration by parts? Could you edit your answer (using the "edit" button) so as to show us your attempt? — paf, Aug 27 '16 at 17:15
Forgot to do so sorry, n and L are just constants and n is a positive integer — LobCity, Aug 27 '16 at 17:36

score 0 · Accepted Answer · answered Aug 27 '16 at 17:26

0

This looks like a Fourier series integral, in which case $L = \pi$ which would simplify the calculations a bit. But in any case, if you integrate by parts twice, the new integrand should be a constant times $x^{2(n-1)}\cos(n\pi x/L)$. This gives you a "reduction" formula. Applying the reduction formula repeatedly should give you a nice pattern. (Especially of $L=\pi$.)

answered Aug 27 '16 at 17:26

B. Goddard

1

Indeed it is, I will try and see what I get, thanks :) – LobCity Aug 27 '16 at 17:38

Evaluating a specific definite integral

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