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Math Help - Integration Help

  1. #1
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    Integration Help

    How do you integrate (x^2+x+1)e^-x?
    Please show all work and explain, thank you.
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  2. #2
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    Quote Originally Posted by maximade View Post
    How do you integrate (x^2+x+1)e^-x?
    Please show all work and explain, thank you.
    Integration by parts, because your integral is a product of two functions.

    In your problem let u= x^2+x+1 and dv=e^{-x}. (The acronym LIATE will help you remember the order for choosing u and dv).
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  3. #3
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    Quote Originally Posted by maximade View Post
    How do you integrate (x^2+x+1)e^-x?
    Please show all work and explain, thank you.
    Use integration by parts.

    \int{u\,dv} = uv - \int{v\,du}.


    In your case, let u = x^2 + x + 1 so that du = 2x + 1

    Let dv = e^{-x} so that v = -e^{-x}.


    So \int{(x^2 + x + 1)e^{-x}\,dx} = -e^{-x}(x^2 + x + 1) - \int{-e^{-x}(2x + 1)\,dx}

     = -e^{-x}(x^2 + x + 1) + \int{e^{-x}(2x + 1)\,dx}.


    Use integration by parts again.

    Let u = 2x + 1 so that du = 2

    Let dv = e^{-x} so that v = -e^{-x}.


    -e^{-x}(x^2 + x + 1) + \int{e^{-x}(2x + 1)\,dx}

     = -e^{-x}(x^2 + x + 1) -e^{-x}(2x + 1) - \int{-2e^{-x}\,dx}

     = -e^{-x}(x^2 + x + 1) - e^{-x}(2x + 1) + 2\int{e^{-x}\,dx}

     = -e^{-x}(x^2 + x + 1) - e^{-x}(2x + 1) - 2e^{-x} + C

     = -e^{-x}(x^2 + x + 1 + 2x + 1 + 2) + C

     = -e^{-x}(x^2 + 3x + 4) + C.
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