# Definite integral involving Log, Sin and evaluates to Zero

I need to evaluate this integral.

$$\int_{- \frac{\pi}{2}} ^{\frac{\pi}{2}} \ln( |1 + 2 \sin{x} |) ~ \mathrm{d} x$$

By using Wolfram, the answer is 0.

How do I solve this algebraically?

My progress -

\begin{align} I = \int _{-\frac{\pi}{2}} ^{\frac{\pi}{2}} \ln |1+2 \sin{x}| \, dx &= \int _0 ^{\frac{\pi}{2}} \ln |1-4 \sin ^2 {x}| \, dx \\\\ &= \int _0 ^{\frac{\pi}{4}} \ln |1-4 \sin ^2 {x}| \, dx + \int _{\frac{\pi}{2}} ^{\frac{\pi}{2}} \ln |1-4 \sin ^2 {x}| \, dx \\\\ &= \int _{0} ^{\frac{\pi}{4}} \ln |1 - 2 \sin{2x}| \, dx + \int _{\frac{\pi}{4}} ^{\frac{\pi}{2}} \ln |1+2 \sin{2x}| \, dx \end{align}

The motivation for the last step is to somehow get similar looking integrals so that I can finally get an expression of the type $$I = k I$$

Thank you for all the comments, I have edited the post to include my progress.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Sep 2, 2021 at 16:07
• Can you solve $1+2\sin x \ge 0$ ? Sep 2, 2021 at 16:08
• @PierreCarre Yes, in the domain of our integral, it would be $x \geq \frac{- \pi}{6}$ Sep 2, 2021 at 16:12
• Welcome to MSE. Your question is getting downvotes (not from me) because it doesn't conform to the site standards. It is expected that in addition to stating the problem, you show your attempts at solving it. What did you try, how far did you get, what are your thoughts, etc? Let me stress that's it is the question that's getting the downvotes, not you. Sep 2, 2021 at 16:16
• @saulspatz Thank you for informing me that, I have now edited the OP. Sep 3, 2021 at 4:58

\begin{align} I = \int _{-\frac{\pi}{2}} ^{\frac{\pi}{2}} \ln |1+2 \sin{x}| \, dx &= \int _0 ^{\frac{\pi}{2}} \ln |1-4 \sin ^2 {x}| \, dx \\\\ &= \int_0^{\frac{\pi}{2}}\ln|1-2\cos{2x}| \,dx \\\\ &= \int_0^{\pi}\frac{1}{2} \ln|1-2\cos{x}| \, dx \\\\ &= \int_0^{\frac{\pi}{2}} \frac{1}{2} \ln|1-4\cos^2{x}| \, dx \\\\ &= \int _0 ^{\frac{\pi}{2}} \frac{1}{2} \ln |1-4 \sin ^2 {x}| \, dx \\\\ &= \frac{I}{2} \end{align}
Hence, we get $$I = 0$$
I think I also need to prove it's convergence as the integrand goes to $$- \infty$$ also. But I don't know how to do that.