A log trig integral \int_0^{\pi/2}\ln\left(1+a\sin^2 x \right) \,dx

A log trig integral \int_0^{\pi/2}\ln\left(1+a\sin^2 x \right) \,dx

August 27, 2022

In this post we will prove the following result

$\begin{align*} \int_0^{\pi/2}\ln\left(1+a\sin^2 x \right) \,dx&=\pi \ln \frac{1+\sqrt{1+a}}{2} \end{align*}$

Click here for the proof

We have relied on the previous established result

$\begin{align*} \sum_{n=1}^\infty \binom{2n}{n}\frac{x^n}{ 2^{2n} n}&=2\ln\left(\frac{2}{1+\sqrt{1-x}} \right) \end{align*}$

Proof here.

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