# Thread: nasty looking but actually easy

1. ## nasty 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.

2. 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.

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

Just thought I would share.
by inspection: $\displaystyle \int ( \sin(x))^{x}\left[ \ln( \sin(x))+x \cot(x)\right]dx = ( \sin x)^x + C$

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

3. 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.

4. 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

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

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