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Integration by Parts x * 5^x

  1. Feb 20, 2013 #1
    1. The problem statement, all variables and given/known data
    integrate by parts.

    Integral: x * 5^x


    2. Relevant equations



    3. The attempt at a solution
    i got to (1/ln5) * 5^x ;; and i'm not sure how to integrate further.
     
  2. jcsd
  3. Feb 20, 2013 #2

    SammyS

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    How about giving a few details regarding how you got that answer & where you are in the process of integration by parts.
     
  4. Feb 20, 2013 #3
    integral (x * 5^x)

    u=x; du=dx
    dv=5^x ; v=(1/ln5)(5^x)

    (x/ln5)5^x - integral ((1/ln5)(5^x) dx)
     
  5. Feb 20, 2013 #4
    Hi whatlifeforme :)

    You have to use the formula:

    [itex]\int f(x)g'(x)dx = f(x)g(x)-\int f'(x)g(x)dx[/itex]

    In this case

    [itex]f(x)= x\implies f'(x)= 1[/itex]

    [itex]g'(x)= 5^{x}= e^{x\ln(5)}\implies g(x)=\frac{e^{x\ln(5)}}{\ln(5)}= \frac{5^x}{\ln(5)} [/itex]


    so [itex]\int f(x)g'(x)dx = f(x)g(x)-\int f'(x)g(x)dx[/itex]

    becomes

    [itex]\int x 5^x dx = x \frac{5^{x}}{\ln(5)}-\int \frac{5^{x}}{\ln(5)}dx[/itex]

    Now you have to solve

    [itex]\int \frac{5^x}{\ln(5)}dx= \frac{1}{\ln(5)}\int 5^xdx[/itex]

    ;)
     
  6. Feb 20, 2013 #5
    so would the final simplified answer be:

    (x/ln5)(5^x) - (5^x/(ln(5)^2)
     
  7. Feb 20, 2013 #6

    SammyS

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    ... plus the constant of integration.

    Yes.

    Check it by differentiating.
     
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