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Math Help - Calculus II integration by parts of exponent functions

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    Calculus II integration by parts of exponent functions

    Why I can not get the original function after I integrate it


    see attachment please

    Calculus II integration by parts of exponent functions-daum_equation_1373505988352.png
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    Re: Calculus II integration by parts of exponent functions

    Quote Originally Posted by mohammedlabeeb View Post
    Why I can not get the original function after I integrate it


    see attachment please

    Click image for larger version. 

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    To start with e^x \cdot e^x = e^{2x} \text{ not } \neq e^{x^2}

    The integral \int e^{x^2}dx cannot be expressed in exact form. Is this a definite integral you left the limits off of perhaps?

    -Dan
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    Re: Calculus II integration by parts of exponent functions

    Topsquark is correct that \displaystyle \begin{align*} \int{e^{x^2}\,dx} \end{align*} can not be expressed in a closed form in terms of the elementary functions. However, it can be expressed as an infinite series.

    \displaystyle \begin{align*} e^X &= \sum_{n = 0}^{\infty}{\frac{X^n}{n!}} \\ e^{x^2} &= \sum_{n = 0}^{\infty}{\frac{\left( x^2 \right) ^n }{n!}} \\ &= \sum_{n = 0}^{\infty}{\frac{x^{2n}}{n!}} \\ \int{e^{x^2}\,dx} &= \int{\sum_{n = 0}^{\infty}{\frac{x^{2n}}{n!}}} \\ &= \sum_{n = 0}^{\infty}{\frac{x^{2n + 1}}{\left( 2n + 1 \right)  n!}} \end{align*}
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