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Math Help - A hard to crack definite integral

  1. #1
    niz
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    A hard to crack definite integral

    Please help me solve the following integral

    ∫ t^2 e^t Cost dt
    0

    I have been hinted that this integral can be solved using Laplace transform but I don't see any e^ -t in the integrand and if I assume e^t = e^-(-t) then the result become infinity. Please help.
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  2. #2
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    Quote Originally Posted by niz View Post
    Please help me solve the following integral

    ∫ t^2 e^t Cost dt
    0

    I have been hinted that this integral can be solved using Laplace transform but I don't see any e^ -t in the integrand and if I assume e^t = e^-(-t) then the result become infinity. Please help.
    The result of that integral is infinity (the t^2 and exponential terms blow up) I can only assume you copied it down wrong, or that it was a misprint and they meant e^-t
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  3. #3
    Eater of Worlds
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    \int t^{2}e^{t}cos(t)dt

    Tabular integration may work well with this.

    +\rightarrow t^{2}\searrow \;\ \;\ \rightarrow e^{t}cos(t)

    -\rightarrow 2t\searrow \;\ \;\ \rightarrow \frac{e^{t}cos(t)}{2}+\frac{e^{t}sin(t)}{2}

    +\rightarrow 2\searrow \;\ \;\ \rightarrow \frac{e^{t}sin(t)}{2}

    -\rightarrow 0 \;\ \;\ \rightarrow \frac{e^{t}sin(t)}{4}-\frac{e^{t}cos(t)}{4}

    Now, add up the signed products of the diagonals.

    \frac{t^{2}e^{t}cos(t)}{2}+\frac{t^{2}e^{t}sin(t)}  {2}-te^{t}sin(t)+\frac{e^{t}sin(t)}{2}-\frac{e^{t}cos(t)}{2}

    Here's the graph.

    The lower limit gives -1/2.

    But, see what happens with the upper limit?. As LostChild stated, if that were e^{-t} then that is another matter.

    If we had \int_{-\infty}^{0}t^{2}e^{t}cos(t)dt, then we would have a defintive solution.
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  4. #4
    niz
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    Yes, I think there was a print mistake in the book. Thank a lot for the help.
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by niz View Post
    Please help me solve the following integral

    ∫ t^2 e^t Cost dt
    0

    I have been hinted that this integral can be solved using Laplace transform but I don't see any e^ -t in the integrand and if I assume e^t = e^-(-t) then the result become infinity. Please help.
    Assuming, as everyone said that this should be \int_0^{\infty}t^2e^{-t}\cos(t)dt we can use the trick that then our integral is equal to \Re\int_0^{\infty}t^2e^{(i-1)t}dt which is a much easier integral to compute.
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