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Homework Help: Can this integral be done using elementary techniques?

  1. Aug 20, 2013 #1
    1. The problem statement, all variables and given/known data

    ## \displaystyle \int \frac{dx}{(x^{2}+a^{2})^{3/2}} ##

    2. Relevant equations

    3. The attempt at a solution
    A substitution does not seem to work. I know that a closed form solution exists however, just curious if it can be done by the standard techniques usually taught in calculus. Thanks!

  2. jcsd
  3. Aug 20, 2013 #2
    The substitution [itex] x=asinh\theta [/itex] seems to work.

    Edit: [itex] x=atan\theta [/itex] also works if you don't mind integrating the secant function.
    Last edited: Aug 20, 2013
  4. Aug 20, 2013 #3
    Actually, you need to integrate cosine after that substitution. It may be the easiest way.
  5. Aug 20, 2013 #4
    You are right of course. Somehow I turned [itex] \frac{1}{sec\theta} [/itex] into [itex] \frac{1}{cos\theta} [/itex]
  6. Aug 20, 2013 #5


    Staff: Mentor

    The presence of x2 + a2 in the denominator makes this integral a candidate for trig substitution. Same goes for the difference of squares, either x2 - a2 or a2 - x2.

    What I do in these situations is draw a right triangle, and label the sides and hypotenuse according to whether I'm dealing with a sum of squares or a difference. If it's a sum of squares, as in this problem, I label the vertical side as x and the horizontal side as a. This means that the hypotenuse is √(x2 + a2). If θ is the angle across from x, then tanθ = x/a, so sec2θdθ = dx/a, or dx = asec2θdθ. After this, replace x and dx in the integral with θ and dθ and integrate.

    A similar analysis can be done for either of the two differences of squares.
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