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Math Help - Integral Question

  1. #1
    Member RedBarchetta's Avatar
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    Integral Question

    Umm, I'm not quite sure how to ask this....

    <br />
\int {x^3 e^{x^2 } } dx = \tfrac{1}<br />
{2}\int {x^2 e^{x^2 } } d(x^2 )<br />

    How do you do that? What's going on here? As in changing dx to d(x^2). Is this a method I haven't been taught?....or do I not fully understand the concept of "with respect to x"? I see that d(x^2) equals 2x giving you the same integral but I need further clarification....

    Thanks.
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  2. #2
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    Opalg's Avatar
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    This is just an alternative notation for making a substitution in an integral. If you replace x^2 by y in the right-hand integral then it becomes \tfrac{1}{2}\!\!\int y e^y\,dy, which exactly what you would get by making the substitution y=x^2 in the left-hand integral.
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  3. #3
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    Quote Originally Posted by RedBarchetta View Post
    Umm, I'm not quite sure how to ask this....

    <br />
\int {x^3 e^{x^2 } } dx = \tfrac{1}<br />
{2}\int {x^2 e^{x^2 } } d(x^2 )<br />

    How do you do that? What's going on here? As in changing dx to d(x^2). Is this a method I haven't been taught?....or do I not fully understand the concept of "with respect to x"? I see that d(x^2) equals 2x giving you the same integral but I need further clarification....

    Thanks.
    d(x^2) = 2 x \, dx. Substitute and you get your original integral.

    Perhaps a more obvious approach for you is to make the substitution

    u = x^2 \Rightarrow \frac{du}{dx} = 2x \Rightarrow dx = \frac{du}{2x}

    in your integral. You get

    \frac{1}{2} \int u e^u \, du which can be solved using integration by parts.
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