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Homework Help: Cal 2 integral / trig substitution

  1. Sep 14, 2011 #1
    1. The problem statement, all variables and given/known data

    I am asked to prove the following statement is correct

    integral (sqrt(a^2+x^2))/x dx = sqrt(a^2+x^2)-a log(a (sqrt(a^2+x^2)+a))+ C

    2. Relevant equations

    x = atanθ
    dx = (asecθ)^2

    tan^2+1 = sec^2

    3. The attempt at a solution

    got down to a (sec^2 θ a(√sec^2)dθ)/atanθ

    I plugged into wolfram and immediately got something involving csc in the steps and im not sure where it came from. Just beginning these trig substitutions in class.
    Last edited: Sep 14, 2011
  2. jcsd
  3. Sep 14, 2011 #2


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    Homework Helper

    x = atan \theta
    dx = a sec^2 \theta d\theta

    so subbing into the integral you get (might want to check the steps)
    \int \frac{sqrt{a^2+x^2}}{x}dx
    = \int \frac{\sqrt{a^2+a^2tan^2 \theta}}{atan\theta} a sec^2 \theta d\theta
    = \int \frac{\sqrt{a^2sec^2\theta}}{tan\theta} sec^2 \theta d\theta
    = \int \frac{a}{cos\theta}\frac{cos\theta}{sin\theta} \frac{1}{cos^2 \theta} d\theta
    = \int a\frac{1}{sin\theta} \frac{1}{cos^2 \theta} d\theta

    now can you make another substitution?
  4. Sep 15, 2011 #3
    would you use

    U= secθ
    dU =sec(θ)tan(θ)
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