Challenge problem by friend is $$\int_0^{\frac{\pi}{2}} \frac{\sin^3{(2x)}}{\ln{\left(\csc{x}\right)}} \mathop{dx}$$
I know you can write $\ln{\left(\csc{x}\right)}=-\ln{\sin{x}}$ and $\sin{(2x)}=2\sin{(x)}\cos{(x)}$. I tried rewriting the integral but then could not go further. Even Wolfram Alpha (https://www.wolframalpha.com/input/?i=integral+of+sin%5E3%282x%29%2F%28log%28csc%28x%29%29%29+dx+from+0+to+pi%2F2) could not get a closed form!? Is it even possible.
Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin^3{(2x)}}{\ln{\left(\csc{x}\right)}} \mathop{dx}$
calculusdefinite integralsindefinite-integralsintegrationreal-analysis
Best Answer
You have the correct idea of expanding $\sin{(2x)}$: $$I=8\int_0^{\frac{\pi}{2}} \frac{\sin^3{x}\cos^3{x}}{\ln{\left(\csc{x}\right)}} \; dx$$ Substituting $u=\ln{\left(\csc{x}\right)}$ will be very helpful: $$I=8\int_{\infty}^0 \frac{\sin^3{x}\cos^3{x}}{u} \cdot \frac{du}{-\cot{x}}$$ $$=8\int_{0}^{\infty} \frac{\sin^4{x}\left(1-\sin^2{x}\right)}{u} \; du$$ $$=8\int_{0}^{\infty} \frac{e^{-4u}\left(1-e^{-2u}\right)}{u} \; du$$ $$=8\int_{0}^{\infty} \frac{e^{-4u}-e^{-6u}}{u} \; du$$ Now, this is a simple application of the Frullani integral: $$8 \cdot \ln{\left(\frac{-6}{-4}\right)}=\boxed{8\ln{\left(\frac{3}{2}\right)}}$$ So, it turns out that this integral does have a closed form expression.