Math Help - How to calculate this integral?

1. How to calculate this integral?

Hey, how are you? I was troubled by calculating $f(a)=\int_0^\pi \log(1+a \cos x)dx.$ Here, $|a|<1$ is real.

I do know we can differentiating under the integral, however, I could not see any more...

2. Re: How to calculate this integral?

That seems like a reasonable approach. Differentiate with respect to a and exchange the derivative and integral. You should get a rational function of cos x which you can integrate. Then you will have to integrate with respect to a and evaluate the constant by looking at a=0.

So what do you get for the derivative of log(1+a cos x) with respect to a?

- Hollywood

3. Re: How to calculate this integral?

In fact, I do the series expansion. It is difficult to integrate the derivative of log(1+a cos x) with respect to a, isn't it?

4. Re: How to calculate this integral?

Yes, I think that would qualify as a difficult integral. But it's not impossible. You can substitute $u=\tan\frac{x}{2}$ to convert any rational combination of trig functions to a rational function of u, which can then be integrated using partial fractions. But then it might not be possible to integrate the result with respect to a.