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Math Help - Integration

  1. #1
    Member srirahulan's Avatar
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    Lightbulb Integration

    This my big problem in my integration problem, \int{e^x \ln x.In this case i choose u= \ln x and \frac{de^x} {dx}=e^x.Then i stuck on this \int{\frac{e^x}{x}}.please give me a solution
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  2. #2
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    Re: Integration

    I agree with your method of integration by parts, so you should end up with \displaystyle \begin{align*} \int{e^x\ln{(x)}\,dx} = e^x\ln{(x)} - \int{\frac{e^x}{x}\,dx} \end{align*}. The second function does not have a solution in terms of the elementary functions, but Wolfram tells us that this is the Exponential Integral.
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  3. #3
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    Re: Integration

    Of course, a series solution could be appropriate...

    \displaystyle \begin{align*} e^x &= \sum_{n = 0}^{\infty}{\frac{x^n}{n!}} \\ \frac{e^x}{x} &= \sum_{n = 0}^{\infty}{\frac{x^{n-1}}{n!}} \\ &= \frac{1}{x} + \sum_{n = 1}^{\infty}{\frac{x^{n-1}}{n!}} \\ \int{\frac{e^x}{x}\,dx} &= \int{\frac{1}{x} + \sum_{n=1}^{\infty}{\frac{x^{n-1}}{n!}}\,dx} \\ &= \ln{|x|} + \sum_{n = 1}^{\infty}{\frac{x^n}{n \cdot n!}} + C \end{align*}
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  4. #4
    Member srirahulan's Avatar
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    Re: Integration

    If there is no other way to approach this problem,like let u=e^x and d[xlnx-x]/dx=lnx.
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  5. #5
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    Re: Integration

    Quote Originally Posted by srirahulan View Post
    If there is no other way to approach this problem,like let u=e^x and d[xlnx-x]/dx=lnx.
    No, that gives you a more difficult integral to try to solve. Not all integrals have closed-form solutions in terms of elementary functions.
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