# Log-sine and log-cosine integrals

#### Random Variable

##### Well-known member
MHB Math Helper
For a few of you, this probably isn't very challenging. But I'm going to post it anyways since I find it interesting.

Show that for $0 \le \theta \le \pi$, $\displaystyle \int_{0}^{\theta} \ln(\sin x) \ dx = - \theta \ln 2 - \frac{1}{2} \sum_{n=1}^{\infty} \frac{\sin (2n \theta)}{n^{2}}$.

Also show that for $0 \le \theta \le \frac{\pi}{2}$, $\displaystyle \int_{0}^{\theta} \ln(\cos x) \ dx = - \theta \ln 2 + \frac{1}{2} \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\sin (2n \theta)}{n^{2}}$.

#### DreamWeaver

##### Well-known member
I quite agree, RV... Very interesting! I'll not answer that one for reasons we both understand, but if it's not entirely impertinent of me - which it might well be - I'd like to propose the following one for you... Something to get you teeth into.

For $$\displaystyle a,\, b,\, c > 0 \in \mathbb{R}$$, $$\displaystyle m\in \mathbb{Z}^+\ge 1$$, and $$\displaystyle 0 < \theta \le \pi/2$$, find the additional conditions on the parameters as well as the closed form solution for:

$$\displaystyle \int_0^{\theta}\log^m(a+b\cos x +c\sin x)\,dx$$