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Math Help - integration by substitution xe^x^2

  1. #1
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    integration by substitution xe^x^2

    Question is to calculate the integral \int\limits_0^2 {xe^{x^2 } dx}  =

    This is what I have done, however looks to be incorrect.

    I let u = e^{x^2 }

    and so \begin{array}{l}<br />
 \frac{{du}}{{dx}} = 2xe^{x^2 }  \\ <br />
 du = 2xe^{x^2 } dx \\ <br />
 \end{array}

    and <br />
\begin{array}{l}<br />
 \frac{1}{2}\int {udu}  \\ <br />
  = \frac{1}{2}\left[ {\frac{{u^2 }}{2}} \right]_0^2  \\ <br />
  = \frac{1}{2}\left[ {\frac{{(e^{x^2 } )^2 }}{2}} \right]_0^2  \\ <br />
  = \frac{1}{2}\left[ {\left( {\frac{{(e^{2^2 } )^2 }}{2}} \right) - \left( {\frac{{(e^{0^2 } )^2 }}{2}} \right)} \right] \\ <br />
  = \frac{1}{2}\left( {\frac{1}{2}e^{16}  - \frac{1}{2}} \right) \\ <br />
 \end{array}<br />
    However answer is given as <br />
\frac{1}{2}\left( {e^4  - 1} \right)<br />

    Can someone tell me the errors of my way. Thanks
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  2. #2
    o_O
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    You don't get \int u du with that substitution.

    Use: u = x^2 \ \Rightarrow \ du = 2x \ dx \iff {\color{red}\frac{1}{2}du} = {\color{red}x \ dx}

    Note that: u(x) = x^2 \ \Rightarrow u(2) = (2)^2 = 4 \ \text{and} \ u(0) = (0)^2 = 0

    So: \int_{0}^{2} {\color{red}x}e^{x^2} {\color{red}dx} = \int_{u(0)}^{u(2)} e^{u} \left({\color{red}\frac{1}{2} du}\right) = \frac{1}{2} \int_{0}^{4} e^u \ du
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