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Math Help - integral for cos(xt)/(1+t) dt

  1. #1
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    integral for cos(xt)/(1+t) dt

    cost(xt)/1+t dt anyone can help
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  2. #2
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    i need to show it with leibnez rule but i cant find how
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  3. #3
    MHF Contributor chisigma's Avatar
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    Lets start evaluating the indefinite integral...

    \varphi (x, t)= \int \frac{e^{x t}}{1+t}\cdot dt (1)

    Setting 1+t=\tau You obtain first the integral...

    \varphi (x,\tau) = e^{-x} \int \frac{e^{x \tau}}{\tau}\cdot d\tau (2)

    ... and now setting x \tau = \xi...

    \varphi (x,\xi)= e^{-x} \int \frac{e^{\xi}}{\xi}\cdot d\xi = e^{-x}\cdot Ei(\xi) + c (3)

    ... where Ei(*) is the socalled 'Integral exponential function'. What we have obtained is that...

    \int \frac{e^{x t}}{1+t}\cdot dt = e^{-x}\cdot Ei [x\cdot (1+t)] + c (4)

    If now we write in (4) i x instead of x we obtain...

    \int \frac{\cos x t}{1+t}\cdot dt = Re \{ e^{-i x}\cdot Ei[i x \cdot (1+t)]\} + c (5)

    Kind regards

    \chi \sigma
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