1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Algebraic Integral

  1. Oct 10, 2013 #1

    utkarshakash

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    [itex] \displaystyle \int_0^{\infty} \dfrac{x^2-1}{x^4+1} dx [/itex]

    2. Relevant equations

    3. The attempt at a solution
    If I separate the integral I get

    [itex] \displaystyle \int_0^{\infty} \dfrac{x^2}{x^4+1} dx - \int_0^{\infty} \dfrac{1}{x^4+1} dx[/itex]

    But these two integrals are itself complicated.
     
  2. jcsd
  3. Oct 11, 2013 #2
    Yes, it is a difficult integral. Here's an approach using partial fractions. I may not have written it quite right -- please check the arithmetic.

    Let's start with ## x^4 + 1## . This can be rewritten as ##x^4 + 2x^2 +1 - 2x^2 = (x^2 + 1)^2 - 2x^2##. You can factor this expression as ##(x^2 +1 - \sqrt 2x)(x^2 + 1 + \sqrt2x)##. You use the partial fraction method to rewrite this as
    ##\frac{x^2-1}{(x^2 +1 - \sqrt 2x)(x^2 + 1 + \sqrt2x)} = \frac{A}{(x^2 +1 - \sqrt 2x)} + \frac{B}{x^2 + 1 + \sqrt2x}## where A and B are real numbers.

    Now add up these fractions exactly as you did in elementary school and set the numerator to ##x^2 - 1##. This will allow you to solve for A and B.

    Now how do you integrate ## \frac{A}{(x^2 +1 - \sqrt 2x)}##? This one you can look up. ##\int \frac{1}{ax^2 + bx + c}dx = \frac{2}{d}arctan \frac{2ax + b}{d}## where d = ##\sqrt{4ac - b^2}##
     
  4. Oct 11, 2013 #3
    Rewrite the given integral as:
    [tex]\int_0^{\infty} \frac{1-1/x^2}{x^2+1/x^2} dx[/tex]

    ##1-1/x^2## is the derivative of ##x+1/x##. Do you see how to proceed from here?
     
  5. Oct 11, 2013 #4
    Very clever, I like it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted