# Differentiation under the integral

1. Nov 20, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
$$f(\alpha) = \int \log(1+ \alpha \cos(x))dx$$

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?

2. Relevant equations

3. The attempt at a solution

2. Nov 20, 2007

### quasar987

no bounds to that integral?

3. Nov 20, 2007

### ehrenfest

Sorry. It goes from 0 to pi.

4. Nov 21, 2007

### 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.

5. Nov 21, 2007

### danni7070

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

6. Nov 21, 2007

### Gib Z

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