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!

Homework Help: Difficult Integration - Apostol Section 6.25 #40

  1. Sep 4, 2011 #1
    1. The problem statement, all variables and given/known data

    Hint: multiply the numerator and denominator by [itex]\sqrt{2-x-x^2}[/itex]

    2. Relevant equations
    This is in the Integration using Partial Fractions section, but the last few have not been using Partial Fractions.

    3. The attempt at a solution
    Well, initially I thought that I would just complete the square on the top and then use a substitution such as [itex]x+\frac{1}{2} = \frac{3}{2}\sin u[/itex], but that became pretty complex. Then I took the author's suggestion and multiplied the numerator and denominator by [itex]\sqrt{2-x-x^2}[/itex]. I split the resulting integral into three, two of which were easy and the first which is still very difficult:

    [itex]\int \frac{2}{x^2 \sqrt{2-x-x^2}} dx[/itex]

    The substitution I mentioned earlier still looks most promising, but leads to this:

    [itex]\frac{9}{4} \int \frac{\cos^2 u}{(\frac{3}{2}\sin u + \frac{1}{2})^2} du[/itex]

    This looks like it needs something like [itex]z=\tan \frac{u}{2}[/itex], but that also becomes extremely tortuous. It leads to a degree 6 polynomial on the bottom, and a degree 4 polynomial on the top which can then be solved with partial fractions, but the resulting equations are quite cumbersome.

    Any other suggestions/hints?
  2. jcsd
  3. Sep 4, 2011 #2


    User Avatar
    Homework Helper

    Try integration by parts:

    [itex]9\int \frac{\cos^2 u}{(3\sin u + 1)^2} du=\int (\frac{3\cos u}{(3\sin u+1)^2}) (3 \cos u) du[/itex]

  4. Sep 4, 2011 #3
    That does look much better, thanks!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook