Differentiation under the integral

[ehrenfest]

Homework Statement

$$f(\alpha) = \int \log(1+ \alpha \cos(x))dx$$

I am supposed to differentiate w.r.t alpha and then integrate to find f(alpha).

My book says that there should be a factor of pi in the answer but I do not get one. Does anyone else?

Homework Equations

The Attempt at a Solution

 

[quasar987]

no bounds to that integral?

 

[ehrenfest]

Sorry. It goes from 0 to pi.

 

[arildno]

Okay, so we differentiate:
$$f'(\alpha)=\int_{0}^{\pi}\frac{\cos(x)}{1+\alpha\cos(x)}dx=\frac{1}{\alpha}\int_{0}^{\pi}(1-\frac{1}{1+\alpha\cos(x)})dx,f(0)=0$$
See if this brings you any further.

The substitution $$u=\tan(\frac{x}{2})$$ might well be helpful.

 

[danni7070]

is log(1+acos(x)) = cos(x)/(1+acos(x)) ?

 

[Gib Z]

No, he was doing what the thread title says. It's the derivative of your integrand with respect to alpha.