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Factoring denominator of an integral

  1. Oct 22, 2011 #1
    1. The problem statement, all variables and given/known data

    Ok, this is a pretty simple integral, but I'm having trouble with the factoring.

    [itex]\int \frac{1}{x^{2}+2}[/itex]

    According to the book, the answer is:

    [itex]\frac{1}{\sqrt{2}} tan^{-1}(\frac{x}{\sqrt{2})}[/itex]

    2. Relevant equations



    3. The attempt at a solution

    So I need to get it in the form of:

    [itex]\int\frac{1}{x^{2}+1}[/itex]

    I did this by factoring out a [itex]\sqrt{2}[/itex]:

    [itex]\frac{1}{\sqrt{2}(\frac{x^{2}}{\sqrt{2}}+\sqrt{2})}[/itex]

    But when you convert the [itex]\frac{x^{2}}{\sqrt{2}}[/itex] to [itex](\frac{x}{\sqrt{2}})^{2}[/itex] the [itex]\sqrt{2}[/itex] on the outside of the factor doesn't cancel out the one being squared. I'm kind of lost here.

    Squaring the bottom term produces a 2 which doesn't cancel with the [itex]\sqrt{2}[/itex] on the outside of the parenthesis, and changing the term to [itex]4\sqrt{2}[/itex] which would square and cancel isn't in the given answer.
     
    Last edited: Oct 22, 2011
  2. jcsd
  3. Oct 22, 2011 #2

    Mark44

    Staff: Mentor

    Don't forget dx! You are consistently omitting it in your integrals. It is crucial in problems like this.

    The simplest way to do this is to use a trig substitution. Draw a right triangle with an acute angle θ. Label the altitude as x and the base as 2. From this we see that tan(θ) = x/2, so 2sec2(θ)dθ = dx.

    Replace all expressions with x and dx in your original integral, and you'll have an easier one to integrate.
     
  4. Oct 22, 2011 #3

    Mark44

    Staff: Mentor

    I should mention that you can factor the denominator and use the method of partial fraction decomposition. x2 + 2 factors into (x + i√2)(x - i√2).
     
  5. Oct 23, 2011 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    If you need [itex]x^2+ [/itex] rather than [itex]x^2+ 2[/itex] you surely don't want [itex]x^2+ \sqrt{2}[/itex]! Don't factor out [itex]\sqrt{2}[/itex], factor out [itex]2[/itex]:
    [tex]\frac{1}{2}\int \frac{dx}{\frac{x^2}{2}+ 1}[/tex]
    Now let [itex]u= x/\sqrt{2}[/itex]. As Mark44 says,don't forget the dx! [itex]x= \sqrt{2}u[/itex] so [itex]dx= \sqrt{2}du[/itex]

     
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