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Math Help - Hard integral from easy function

  1. #1
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    Hard integral from easy function

    Hi!

    I want to calculate the integral from x^x dx or x^(1/x) dx, but I don't know how I can start it. I would appreciate for any help or maybe if someone have a book with solved integrals like this two, then he could give a title or link to this.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by fliker View Post
    Hi!

    I want to calculate the integral from x^x dx or x^(1/x) dx, but I don't know how I can start it. I would appreciate for any help or maybe if someone have a book with solved integrals like this two, then he could give a title or link to this.
    You can do \int_0^1 x^x\text{ }dx but there is no anti-derivative.
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    How calculate definite?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by fliker View Post
    How calculate definite?
    \int_0^1 x^x\text{ }dx=\int_0^1 e^{x\ln(x)}\text{ }dx=\int_0^1\sum_{n=1}^{\infty}\frac{x^n\ln^n(x)}{  n!} =\sum_{n=1}^{\infty}\int_0^1 \frac{x^n\ln^n(x)}{n!}=... etc.
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