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Math Help - Integral of t^3*(e^4t^2) and t^2*(e^4t)

  1. #1
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    Integral of t^3*(e^4t^2) and t^2*(e^4t)

    Been doing plenty of practice of late just to get back in the swing of things. Could someone please help me out with the solutions to the above?

    I think they may involve integration by parts twice, which sounds ugly.

    I can't find any similar examples in notes or text book...


    Regards.
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  2. #2
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    Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

    Quote Originally Posted by paulbk108 View Post
    Been doing plenty of practice of late just to get back in the swing of things. Could someone please help me out with the solutions to the above?

    I think they may involve integration by parts twice, which sounds ugly.

    I can't find any similar examples in notes or text book...


    Regards.
    For the first one take: u=t^2 and dv=t\cdot e^{4t^2} dt

    For the second one take  u=t^2 and dv=e^{4t}
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  3. #3
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    Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

    \int t^3(e^{4t^2}) \mathrm{d}t
    Substitute x=t^2.

    \int t^3(e^{4t^2})\ \mathrm{d}t=\int \frac{x \cdot e^{4x}}{2}\ \mathrm{d}x

    ={1 \over 2}\left[ \frac{e^{4x}x}{4}-\frac{1}{4}\int e^{4x} dx\right]+C={1 \over 2}\left[ \frac{e^{4x}x}{4}-\frac{1}{16}e^{4x}\right]+C
    Last edited by sbhatnagar; December 6th 2011 at 02:29 AM.
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  4. #4
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    Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

    One of you folks mind completing one of them?

    Once I get rolling I will be fine...


    Regards.
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  5. #5
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    Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

    sbhatnagar,

    Can you please explain where the t^3 went? You substituted x = t^2...?


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  6. #6
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    Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

    I substituted x=t^2

    \frac{dx}{dt}=2t \iff dt=\frac{dx}{2t}

    \int t^3{e^{4t^2}} dt=\int \frac{t^3(e^{4x})}{2t} dx=\int\frac{t^2(e^{4x})}{2} dx=\int \frac{x \cdot e^{4x}}{2} dx
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  7. #7
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    Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

    Quote Originally Posted by paulbk108 View Post
    One of you folks mind completing one of them?

    Once I get rolling I will be fine...


    Regards.
    Math Help Forum is not a homework service. You are expected to show some effort, particularly after some help has been given. Show us where you are stuck, and we'll help you get unstuck. But we are not here simply to do your homework for you.
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