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Math Help - Attempt to solve confusing integral $x^3 *e^[x^2] * dx (See attached scanned paper)

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    Attempt to solve confusing integral $x^3 *e^[x^2] * dx (See attached scanned paper)

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    Quote Originally Posted by Riazy View Post
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    Put \displaystyle{u = x^2\,; \;u'=2x\,,\,\,v'=xe^{x^2}\,;\;v=\frac{1}{2}e^{x^2}  } , so integrating by parts:

    \displaystyle{\int x^3e^{x^2} dx=\int x^2(xe^{x^2})dx=\frac{1}{2}x^2e^{x^2}-\int xe^{x^2}dx=\frac{e^{x^2}}{2}\left(x^2-1)+C .

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    \displaystyle \int{x^3e^{x^2}\,dx} = \int{\frac{1}{2}x^2\cdot 2x\,e^{x^2}\,dx}.

    Now use integration by parts with \displaystyle u = \frac{1}{2}x^2 \implies du = x\,dx and \displaystyle dv = 2x\,e^{x^2}\,dx \implies v = e^{x^2} and the integral becomes

    \displaystyle \frac{1}{2}x^2e^{x^2} - \int{x\,e^{x^2}\,dx}

    \displaystyle = \frac{1}{2}x^2e^{x^2} - \frac{1}{2}\int{2x\,e^{x^2}\,dx}

    \displaystyle = \frac{1}{2}x^2e^{x^2} - \frac{1}{2}e^{x^2} + C.
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