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Integral I'm not able to solve

  1. Jun 11, 2014 #1
    Hello! I'm having some troubles with that integral:

    ## \int_0^{k} \frac{x^{\alpha}}{1 + \beta x} dx##

    I've tried to think a lot on this but I've no idea how to solve it, so I hope someone could help me. Thank you!
    Last edited: Jun 11, 2014
  2. jcsd
  3. Jun 11, 2014 #2
    Assuming 'k' is a variable, you could write f(k)= ## \int_0^{k} \frac{x^{\alpha}}{1 + \beta x} dx##. Try finding f'(k) and solve the problem.
  4. Jun 11, 2014 #3
    You can solve this with the hypergeometric function:

    I don't think you can solve it without using such a function.
  5. Jun 11, 2014 #4


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    f'(k) is, of course, ##\frac{k^{\alpha}}{1+ \beta k}## but I don't see how that helps find f(k).
  6. Jun 11, 2014 #5

    D H

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    How it helps is that one can express the integral as an infinite series. Obviously, f(0)=0. All one needs to form the infinite series is f'(k), f''(k), and so on.
  7. Jun 11, 2014 #6
    Yeah and this method requires that we know the values of a and b.
  8. Jun 11, 2014 #7

    D H

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    No, it doesn't.
  9. Jun 11, 2014 #8
    The integral can be easily evaluated if ##k\rightarrow \infty## and has a nice result.
  10. Jun 13, 2014 #9


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    Well, it's also easy to evaluate the integral if either α→0 or β→0. But the OP asked for a general solution rather than a solution for k, α, or β approaching some limit.

    Besides, as k→∞, the integral is only defined when α is between -1 and 0, right? For positive (or zero) α, the integral diverges as k→∞. For α<-1, there is a problem integrating in the region near x=0.
  11. Jun 13, 2014 #10
    Of course there are restrictions. Another restriction is that ##\beta >0##. With these, the result of the definite integral is:
    $$\frac{-1}{\beta^{\alpha+1}}\frac{\pi}{\sin(\pi \alpha)}$$
    Last edited: Jun 13, 2014
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