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Math Help - Integral problem

  1. #1
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    Integral problem

    \int_0^{00}\ {sinx /x} dx\
    the ans is {pi/2}
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  2. #2
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    Krizalid's Avatar
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    i know a solution that covers double integration, but, have you covered double integrals so that you can get my solution?
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    i know a solution that covers double integration, but, have you covered double integrals so that you can get my solution?
    where is ur solution?
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  4. #4
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    Krizalid's Avatar
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    okay first we integrate by parts to get:

    \int_{0}^{\infty }{\frac{\sin x}{x}\,dx}=\int_{0}^{\infty }{\frac{1-\cos x}{x^{2}}\,dx}, now on the last integral we use the fact that \frac1{x^2}=\int_0^\infty te^{-tx}\,dt so the last integral becomes \int_{0}^{\infty }{\int_{0}^{\infty }{\left( te^{-tx}-te^{-tx}\cos x \right)\,dt}\,dx}, now since f(x,t)\ge0, by using Tonelli, we justify the change of integration order, so this leads to compute \int_{0}^{\infty }{\left( 1-\frac{t^{2}}{t^{2}+1} \right)\,dt}=\frac{\pi }{2}, and we are done.
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