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Math Help - integrate x^(-2/3)*ln(5x)

  1. #1
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    Lightbulb integrate x^(-2/3)*ln(5x)

    Hi all.I have been trying to use integration by parts for this question.

    I have got as far as 3x^(1/3)*Ln(5x)-3 (int)x^(1/3)*1/5x.

    Im a little stuck on the last bit of integration, i.e, (int)x^(1/3)*1/5x.

    I cant see if their is a cancellation to simplify the last bit.

    Am i using the correct rule? Can anyone help????

    Thanks.
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  2. #2
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    Re: integrate x^(-2/3)*ln(5x)

    The derivative of \ln{5x} is not \frac{1}{5x}.

    Either way, surely you know what \frac{x^n}{x} is?
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  3. #3
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    Re: integrate x^(-2/3)*ln(5x)

    Let y = ln u
    u = 5x
    du/dx = 5
    dy/du = 1 / u
    dy/dx = (dy/du) X (du/dx)
    dy/dx = (1/u) X 5
    dy/dx = (1 / 5x) X 5
    dy/dx = 1 / x
    Im failing to see where to use this.

    x^n/x = x^n-1

    Thanks for your reply.
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  4. #4
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    Re: integrate x^(-2/3)*ln(5x)

    Quote Originally Posted by bignaughtydog View Post
    Hi all.I have been trying to use integration by parts for this question.

    I have got as far as 3x^(1/3)*Ln(5x)-3 (int)x^(1/3)*1/5x.

    Im a little stuck on the last bit of integration, i.e, (int)x^(1/3)*1/5x.

    I cant see if their is a cancellation to simplify the last bit.

    Am i using the correct rule? Can anyone help????

    Thanks.
    Using integration by parts: \displaystyle \begin{align*} \int{u\,dv} = u\,v - \int{v\,du} \end{align*} with \displaystyle \begin{align*} u = \ln{(5x)} \implies du = \frac{1}{x}\,dx \end{align*} and \displaystyle \begin{align*} dv = x^{-\frac{2}{3}}\,dx \implies v = 3x^{\frac{1}{3}} \end{align*} we have

    \displaystyle \begin{align*} \int{x^{-\frac{2}{3}}\ln{(5x)}\,dx} &= 3x^{\frac{1}{3}}\ln{(5x)} - \int{3x^{\frac{1}{3}}\cdot \frac{1}{x}\,dx} \\ &= 3x^{\frac{1}{3}}\ln{(5x)} - \int{3x^{-\frac{2}{3}}\,dx} \\ &= 3x^{\frac{1}{3}}\ln{(5x)} - 9x^{\frac{1}{3}} + C \end{align*}
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  5. #5
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    Re: integrate x^(-2/3)*ln(5x)

    Quote Originally Posted by bignaughtydog View Post
    Hi all.I have been trying to use integration by parts for this question.

    I have got as far as 3x^(1/3)*Ln(5x)-3 (int)x^(1/3)*1/5x.

    Im a little stuck on the last bit of integration, i.e, (int)x^(1/3)*1/5x.

    I cant see if their is a cancellation to simplify the last bit.

    Am i using the correct rule? Can anyone help????

    Thanks.
    WA , click "show steps"
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  6. #6
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    Re: integrate x^(-2/3)*ln(5x)

    Thank you thank you priceps and prove it.That was the answer i got after finding the derivative of ln(5x).Ridley did give me a nudge in that direction also.Thanks for that Ridley also.....
    Last edited by bignaughtydog; March 25th 2012 at 04:22 AM.
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