Asymptotics for a trigonometric integral

1. Oct 26, 2013

jozko.slaninka

I have become stuck while trying to evaluate the following trigonometric integral:
$\int_0^{\pi} (A + B \sin x)^n dx.$
First, I have tried to find a recurrence with respect to n, from which the closed-form solution could be calculated. However, I have failed to do this. Similarly, my effort to solve the integral using the binomial theorem seems to be useless, since this leads to the sum that I find relatively hard to evaluate.

So my next thoughts have been to find an asymptotic approximation of this integral, for n tending to infinity. However, I have found that I am unable to do this with my very basic background in asymptotic analysis.

Thus, my question is: could anybody suggest me how to achieve the above mentioned asymptotic approximation (I don't believe in a reasonable closed-form solution of this integral, so I am not asking for any)?