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Math Help - integrate x/(1+cos(x)^2) from 0 to pi

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    integrate x/(1+cos(x)^2) from 0 to pi

    integrate x/(1+cos(x)^2) from 0 to pi
    I thought integration by parts udv = uv - vdu, so x is u and 1/(1+cos(x)^2) is v. But I have trouble finding the trick to integrate v. Thanks for any help or insight
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    Re: integrate x/(1+cos(x)^2) from 0 to pi

    You'll need a numerical integration method to solve this integral.
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    Re: integrate x/(1+cos(x)^2) from 0 to pi

    Is it (cos(x))^2 or cos(x^2) ?
    Is it a school work ? If YES, what exactly is the wording ?
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    Re: integrate x/(1+cos(x)^2) from 0 to pi

    (cos(x))^2

    show the integral x/(1+(cos(x))^2) = (pi^2)/2sqrt(2)
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    Re: integrate x/(1+cos(x)^2) from 0 to pi

    Quote Originally Posted by phigirl View Post
    (cos(x))^2
    show the integral x/(1+(cos(x))^2) = (pi^2)/2sqrt(2)
    OK. That is clear.
    You can solve it thanks to integration by parts with u=x and v(x)=primitive of 1/(1+(cos(x))^2)
    A primitive of 1/(1+(cos(x)^2)) is v(x)=(1/sqrt(2))*arctan(tan(x)/sqrt(2))
    u'=1. So you should have to integrate u'*v=v, wich is very difficult (involving special functions).
    But, you don't need to do it explicitly. Since the function v(x) is periodic, one can see that its definite integal from x=0 to x=pi is equal to 0.
    So, only x*v(x) is remaining. But there is a discontinuity at x=pi/2. Since v' is even then x*v is also even, the definite integral is :
    2*(x*v(x) at x=pi/2) = 2*(pi/2)*(1/sqrt(2))*arctan(tan(pi/2)/sqrt(2)) = 2*(pi/2)*(1/sqrt(2)*(pi/2) = pi²/(2*sqrt(2))
    Last edited by JJacquelin; October 24th 2012 at 01:05 PM.
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