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Math Help - Flawed Knowledge of Reduction Formulas

  1. #1
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    Flawed Knowledge of Reduction Formulas

    Any explanation of how to evaluate this problem through the use of reduction formulas would be greatly appreciated.

    Evaluate: \int(x^nln(x)dx) where x\not=-1

    I know that one must use integration by parts ( \int(udv)=uv-\int vdu)
    As it stands right now I have u=x^n \rightarrow du=nx^{n-1}dx
    Along with dv=ln(x)dx \rightarrow v=xln(x)-x+C

    I substitute the above values into the equation for integration by parts and I do not know how to continue from there. Any advice would be appreciated.
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  2. #2
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    Quote Originally Posted by LithiumPython View Post
    Any explanation of how to evaluate this problem through the use of reduction formulas would be greatly appreciated.

    Evaluate: \int(x^nln(x)dx) where x\not=-1

    I know that one must use integration by parts ( \int(udv)=uv-\int vdu)
    As it stands right now I have u=x^n \rightarrow du=nx^{n-1}dx
    Along with dv=ln(x)dx \rightarrow v=xln(x)-x+C

    I substitute the above values into the equation for integration by parts and I do not know how to continue from there. Any advice would be appreciated.

    It looks pretty straightforward and what you did is right (without the integration constant, of course) , so:

    \int x^n\ln x\,dx= x^{n+1}\ln x - x^{n+1}-n\!\!\int x^n\ln x\,dx+n\!\!\int x^n\,dx\,\Longrightarrow (n+1)\!\!\int x^n\ln<br />
x\,dx=x^{n+1}\left(\ln x-1+\frac{n}{n+1}\right) + C ...

    Tonio
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