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Math Help - Weird Integral

  1. #1
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    Weird Integral

    I am tutoring a student in Calculus 2 and cannot seem to get the following 2 problems (it's been a while since I took this course)

    1. Find the indefinite integral of dx/(x(x^2+4)^2)

    2. Find dy/dx of y=ln(x^2+y^2)
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  2. #2
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    Quote Originally Posted by rainbowkaine View Post

    2. Find dy/dx of y=ln(x^2+y^2)
    Use implicit differentiation.

    y' = \left( \frac{1}{x^2+y^2} \right) \left(2x+2y*y' \right)

    From here, expand the right hand side and then collect all terms containing y' onto one side of the equation, factor out y' and solve algebraically.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by rainbowkaine View Post
    1. Find the indefinite integral of dx/(x(x^2+4)^2)
    hard way: do a trig substitution of x = 2 \tan \theta

    an easier way: note that we have this integral (by multiplying by x/x): \int \frac {x}{x^2(x^2 + 4){\color{red}^2}}~dx

    now, let u = x^2 + 4 \implies \boxed{x^2 = u - 4}

    then du = 2x~dx

    \Rightarrow \frac 12 ~du = x~dx

    So our integral becomes:

    \frac 12 \int \frac 1{u^2(u - 4)}~du

    which is relatively easy to do via partial fractions
    Last edited by mr fantastic; April 9th 2009 at 11:20 PM. Reason: Fixed an obvious typo (red)
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