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Homework Help: Integrating the natural log

  1. Feb 1, 2009 #1
    1. The problem statement, all variables and given/known data
    How would I begin to integrate ln(t+1) from 0 to e^2x?


    2. Relevant equations
    d/dx[log base a of u]=1/(lna)u du/dx

    Can the original equation be manipulated to use this derivative?


    3. The attempt at a solution
    Not sure where to start.
     
  2. jcsd
  3. Feb 1, 2009 #2

    Dick

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    Homework Helper

    It's the sort of problem where you could actually guess the antiderivative, but if you can't, integrate by parts.
     
  4. Feb 7, 2009 #3
    It's an integration by part question. First, use a substitution to get it to one variable instead of a polynomial in the logarithm.
    w=t+1
    dw=dt
    Substitute (I'm going to revert the limits in the end to the original variable, so you know):
    Integrate of ln(w)dw
    Let:
    u=ln(w) and dv=dw
    du=(1/w)*dw and v=w

    w*ln(w)-Int(w*(1/w)*dw)
    w*ln(w)-Int(dw)
    w*ln(w)-w
    (t+1)*[ln(t+1)-1]
    now evaluate at your endpoints:
    {(e^2x+1)*[ln(e^2x+1)-1]}+{1}
    You can probably simplify this some more, but there it is.
     
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