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Math Help - Improper Integral Halp!

  1. #1
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    Improper Integral Halp!

    I can't figure this one out: Integral from 0 to infinity of xarctanxdx/(1+x^2)^2

    I know about lim(T--> Infinity) etc. etc., but I can't figure out the actual integral itself. It appears to be a d/dx arctanx = 1/1+x^2 situation, but I can't seem to get there. I can do it very long and slowly through Integration by parts..

    1) u = x^2, du = 2x.

    =1/2 (1/(1+u)^2)

    and then integrate by parts, but it takes forever and seems wrong.

    Any solutions?
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Sprintz View Post
    I can't figure this one out: Integral from 0 to infinity of xarctanxdx/(1+x^2)^2

    I know about lim(T--> Infinity) etc. etc., but I can't figure out the actual integral itself. It appears to be a d/dx arctanx = 1/1+x^2 situation, but I can't seem to get there. I can do it very long and slowly through Integration by parts..

    1) u = x^2, du = 2x.

    =1/2 (1/(1+u)^2)

    and then integrate by parts, but it takes forever and seems wrong.

    Any solutions?
    ok

    \int \frac{x \tan ^{-1} x }{(x^2+1)^2}dx

    let

    u = \tan ^{-1} x

    \tan u = x \Rightarrow \sec ^2 u du = dx but \sec u = \sqrt{x^2 +1 } so


    \int \frac{(u)\tan u \sec ^2 u }{\sec ^4 u } du

    \int \frac{(u) \tan u }{\sec ^2 u} du

    by parts dv = tan u / sec^2 u , w =u so dw =du

    v =\int \frac{\tan u }{\sec ^2 u } du

    let s = sec u --------> ds = sec u tan u du so

    v = \int \frac{1}{s^3} ds \Rightarrow v = -\frac{s^{-2}}{2} \Rightarrow v = -\frac{\cos ^2 u}{2} ok

    return to our integral

    \int \frac{u \tan u }{\sec ^2 u } du by parts as I said

    so

    -\frac{u\cos ^2 u}{2} + \int \frac{\cos ^2 u}{2} du \Rightarrow -\frac{u\cos ^2 u}{2} +\frac{\sin 2u }{8} + \frac{u}{4}

    but  u = \tan ^{-1} x

    so \int \frac{x \tan ^{-1} x }{(x^2+1)^2) } dx = -\frac{\tan ^{-1}x (\cos ^2(\tan ^{-1}x))}{2}+\frac{\sin 2(\tan ^{-1}x)}{8}+ \frac{\tan ^{-1} x}{4}

    the rest for you I find the integral

    is there anything not clear ?
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