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Math Help - Integration - looking for closed form

  1. #1
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    Integration - looking for closed form

    For the following two integrals,

    1. \int_0^1 \frac{1}{1-x^b} dx

    2. \int_0^1 \frac{x^{ab}}{1-x^b} dx

    is there any closed forms in terms of trigonometric functions ???

    thanks for helping
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  2. #2
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    Quote Originally Posted by graticcio View Post
    For the following two integrals,

    1. \int_0^1 \frac{1}{1-x^b} dx

    2. \int_0^1 \frac{x^{ab}}{1-x^b} dx

    is there any closed forms in terms of trigonometric functions ???

    thanks for helping
    If you give values to a and b and run the result through the Wolfram integrator, I think you'll see the difficulty if not impossibility of a closed form solution in terms of trig functions.

    Eg.
    Attached Thumbnails Attached Thumbnails Integration - looking for closed form-msp.gif  
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  3. #3
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    yes, but there appears to be a pattern...
    some arctan and some log...

    can we use any trigonometric substitution to evaluate?

    thanks
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  4. #4
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    Quote Originally Posted by graticcio View Post
    yes, but there appears to be a pattern...
    some arctan and some log...
    [snip]
    thanks
    Yes, that's probably due to the underlying hypergeometric function .....
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  5. #5
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    so what is the relationship between hypergeometric function and trigonometric function in this case ???

    or how to represent the hypergeometric function here by trigonometric functions?

    thanks
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  6. #6
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    Quote Originally Posted by graticcio View Post
    1. \int_0^1 \frac{1}{1-x^b} dx

    2. \int_0^1 \frac{x^{ab}}{1-x^b} dx
    I think those functions don't converge on the bounds 0 to 1. So the integral doesn't exist.
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  7. #7
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    they converge with x<1
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  8. #8
    Super Member wingless's Avatar
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    Quote Originally Posted by graticcio View Post
    they converge with x<1
    On second thought, you're right..
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