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Math Help - nasty looking but actually easy

  1. #1
    Eater of Worlds
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    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.

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

    Just thought I would share.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by galactus View Post
    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
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  3. #3
    Eater of Worlds
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    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.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by galactus View Post
    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
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  5. #5
    Math Engineering Student
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    Yes, one would've studied first (\sin x)^x=e^{x\ln(\sin x)}, and the derivative seems clear from there.
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  6. #6
    MHF Contributor Mathstud28's Avatar
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    Same concept with \int\frac{\ln(x)}{(1+\ln(x))^2}~dx
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