Compute the integral $\int_0^\pi \frac{\ln(1+\cos(t))}{\cos(t)}dt$ using a double integral.

Question

Consider the integral $$K=\int_0^\pi\frac{\ln(1+\cos(t))dt}{\cos(t)}$$

1. Prove that $$K=\int\int_{\Delta}\frac{dt\,dx}{1+x\cos(t)}$$ where $ \Delta=[0,\pi]\times[0,1)$.
2. What is the value of $K ?$.

I answered the first question by differentiating the function $$F:x\mapsto \int_0^\pi\frac{\ln(1+x\cos(t))dt}{\cos(t)}$$

but i couldn't find the value of $ K$ Any help or idea will be appreciate. Thanks in advance.

Answer 1

Evaluate $K=\int_0^1dx\int_0^\pi\frac{dt}{1+x\cos t}$ by Fubini's theorem with $u=\tan(t/2)$ viz.$$\int_0^\pi\frac{dt}{1+x\cos t}=\int_0^\infty\frac{2du}{1+x+(1-x)u^2}=\frac{\pi}{\sqrt{1-x^2}}$$so$$K=\int_0^1\frac{\pi dx}{\sqrt{1-x^2}}=\tfrac12\pi^2.$$