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Tricky integral

  1. Jul 12, 2011 #1
    1. The problem statement, all variables and given/known data

    indefinite integral: dx/(x*sqrt(9+16x^2))

    2. Relevant equations

    Trig. Substitutions or parts??

    3. The attempt at a solution

    I tried using integration by parts but its got pretty messy....it also resembles a tan trig substitution, but it's within a square root. I'm stumped and can't figure out where to start....

    Can anyone help?

  2. jcsd
  3. Jul 12, 2011 #2


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    Don't use integration by parts. Rewrite the problem as
    [tex]\int\frac{xdx}{x^2\sqrt{9+ 16x^2}}[/tex]
    and let [itex]u= 9+ 16x^2[/itex].
    Last edited by a moderator: Jul 13, 2011
  4. Jul 12, 2011 #3


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    Factor a 3 out of [itex]\sqrt{9 +16x^2}[/itex]

    [tex]\displaystyle \sqrt{9 +16x^2}=\sqrt{9\left(1+\frac{16x^2}{9}\right)}=3 \sqrt{1+\frac{16x^2}{9}}=3 \sqrt{1+\left( \frac{4x}{3} \right)^2 }[/tex]

    We know that 1+tan2(θ) = sec2(θ) , so let 4x/3 = tan(θ), (4/3)dx = sec2(θ)dθ.

    (I'm slow at typing, so HallsofIvy responded while I was typing. He usually has better ideas than I do. Good luck!)
    Last edited: Jul 12, 2011
  5. Jul 12, 2011 #4
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