Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You can solve this with the hypergeometric function:
    http://www.wolframalpha.com/input/?i=int+x^a/(1++b*x)dx
    http://en.wikipedia.org/wiki/Hypergeometric_function

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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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

    User Avatar
    Staff Emeritus
    Science Advisor

    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

    User Avatar
    Staff Emeritus
    Science Advisor

    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

    Redbelly98

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integral I'm not able to solve
  1. Solve this integral (Replies: 5)

Loading...