# Thread: Calculus II integration by parts of exponent functions

1. ## Calculus II integration by parts of exponent functions

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

2. ## Re: Calculus II integration by parts of exponent functions

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

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

3. ## 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*}