I ran across this integral that looks horrendous, but is actually quite simple. I am sure a lot of you will see it right off, but it is kind of fun.

$\displaystyle \int sin(x)^{x}\left(ln(sin(x))+xcot(x)\right)dx$

Just thought I would share.

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- Jan 5th 2009, 03:46 PMgalactusnasty looking but actually easy
I ran across this integral that looks horrendous, but is actually quite simple. I am sure a lot of you will see it right off, but it is kind of fun.

$\displaystyle \int sin(x)^{x}\left(ln(sin(x))+xcot(x)\right)dx$

Just thought I would share. - Jan 5th 2009, 03:52 PMJhevon
- Jan 5th 2009, 04:28 PMgalactus
Yes, of course, I knew you would see it. I just thought it was cool. I am sure we could say that about a lot of them. Note that it includes ln(sin(x)) and xcot(x). Two famous non-elementary integrals having been addressed here on MHF.

- Jan 5th 2009, 04:30 PMJhevon
- Jan 5th 2009, 06:02 PMKrizalid
Yes, one would've studied first $\displaystyle (\sin x)^x=e^{x\ln(\sin x)},$ and the derivative seems clear from there.

- Jan 5th 2009, 06:17 PMMathstud28
Same concept with $\displaystyle \int\frac{\ln(x)}{(1+\ln(x))^2}~dx$ :D