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Integration By Parts

  1. Feb 3, 2014 #1
    1. The problem statement, all variables and given/known data
    ∫cosx(lnsinx)dx


    2. Relevant equations



    3. The attempt at a solution
    u=lnsinx dv=cosxdx
    du=cosx/sinx dx v=sinx

    =(lnsinx)(sinx)-∫(sinx)(cosx/sinx)dx
    =(lnsinx)(sinx)-(sinx)+C

    I thought that I did this correctly, but my teacher said that u should equal sinx. Why would u not equal lnsinx?
     
  2. jcsd
  3. Feb 3, 2014 #2
    If you differentiate, you get the original function, so you did it correctly. I think your teacher was saying u should equal sinx if you're doing a simple u-sub to integrate, since that method works as well. You couldn't have u be sinx for an integration by parts, since it isn't a complete term in the integrand -- it's only part of the natural log.
     
  4. Feb 3, 2014 #3

    Dick

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    You did do it correctly. I think your teacher might be suggesting you do a u-substitution first and then integrate log(u) by parts. I think that's actually a little more complicated, not easier.
     
  5. Feb 3, 2014 #4
    Ohh ok! I didn't know you could use u substitution on that one. Thanks for clearing that up :)
     
  6. Feb 3, 2014 #5

    Dick

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    You can do a u substitution, but then you are left with log(u), which you then need to integrate by parts. Unless you've memorized the integral of log(u). I think your way is better.
     
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