nasty looking but actually easy

**galactus** · January 5th 2009, 03:46 PM

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.

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

Just thought I would share.

**Jhevon** · January 5th 2009, 03:52 PM

Originally Posted by galactus

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.

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

Just thought I would share.

by inspection: $\int ( \sin(x))^{x}\left[ \ln( \sin(x))+x \cot(x)\right]dx = ( \sin x)^x + C$

i just had the urge of checking what the derivative of $(\sin x)^x$ was. because of that log sitting there

**galactus** · January 5th 2009, 04:28 PM

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.

**Jhevon** · January 5th 2009, 04:30 PM

Originally Posted by galactus

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.

indeed. i probably only saw it because you said it was really easy though. because you said that, i looked for ways to find the integral without doing anything difficult. looking to reverse some derivative seemed obvious after that

**Krizalid** · January 5th 2009, 06:02 PM

Yes, one would've studied first $(\sin x)^x=e^{x\ln(\sin x)},$ and the derivative seems clear from there.

**Mathstud28** · January 5th 2009, 06:17 PM

Same concept with $\int\frac{\ln(x)}{(1+\ln(x))^2}~dx$

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