# Exponential function integral with Sin as its index

1. Oct 30, 2011

### Wu Xiaobin

Integral descriptive:
$$\int _{0}^{\pi }\! \left( a+{\it k1}\,\sin \left( \theta+{\it beta1} \right) \right) {e}^{{\it k2}\,\sin \left( \theta+{\it beta2} \right) }{d\theta}$$
I think if beta1 and beta2 come to zero,i may be a typical integral.However,in this integral,beta1 and beta2 are constants but not zero anymore.
If i expand exponential in to series,I will find that

$$\sum _{k=0}^{\infty }{\frac {{{\it k2}}^{k} \left( \theta+{\it beta2} \right) ^{k}}{k!}}$$
However I have difficulty in integrating out the following integral:
$$\int _{0}^{\pi }\! \left( \sin \left( \theta+{\it beta1} \right) \right) ^{k}\sin \left( \theta+{\it beta2} \right) {d\theta}$$
So if any of you have ever met this kind of integral,please try to help me.It is really important to my research.
And in the end,I hope there would be an analytical expression but if it doesn't exist,series form will be ok then.But i would wonder how can i evaluate it in math software like mathematica or maple or matlab due to the convergence problem.