1. Not finding help here? Sign up for a free 30min 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!

Integration by Partial Fractions Help

  1. Mar 18, 2015 #1
    1. The problem statement, all variables and given/known data
    ∫ [x^(3)+4] / [x^(2)+4] dx

    2. Relevant equations
    N/A

    3. The attempt at a solution
    I know that the fraction is improper, so I used long division to rewrite it as x+(-4x+4)/[x^(2)+4].
    Given the form S(x)+R(x)/Q(x), Q(x) is a distinct irreducible quadratic factor [x^(2)+4].
    I used the rule ax^2+bx+c ⇒ (Ax+B)/(ax^2+bx+c) to rewrite it as (Ax+B)/[x^(2)+4].
    I then solved for A and B and got A=-4 and B=4.
    I am now trying to solve ∫ [ x + (-4x+4)/(x^(2)+4) ] dx
    I know that ∫x=(1/2)x^2, but I am stuck with integrating (-4x+4)/[x^(2)+4].
    (I tried u-substitution and that didn't work.)
     
  2. jcsd
  3. Mar 18, 2015 #2

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Hello StrangeCharm. Welcome to PF !

    Split ##\displaystyle \frac{-4x+4}{x^2+4}## into two fractions:

    ##\displaystyle \frac{-4x}{x^2+4} + \frac{4}{x^2+4}##

    The integral of one of them can be done via u-substitution. The other is a fairly well known integral.
     
  4. Mar 18, 2015 #3
    Thanks for the tip!
     
  5. Mar 19, 2015 #4
    I'm almost done with the problem but am having trouble with integrating 4/[x^(2)+4]. It looks like integrating 1/(1+x^2)dx and getting arctan(x); however, I'm not sure how to simplify the fraction into a similar form.
    Also, I'm wondering whether it is valid to make ∫4/(4+x^2)dx ⇒ ∫[2/(2+x)]^2dx and using u-substitution with u=x+2. I'm not sure because doing this would get a different result from the method above.
     
  6. Mar 19, 2015 #5

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Is (2 + x)2 = 4 + x2 ? No.

    Use u substitution: u = 2x .
     
  7. Mar 19, 2015 #6
    Okay, right... I just realized that. Thanks for the help!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted