How does one integrate $\displaystyle e^{x^2}$ using spherical coordinates?
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I don't understand the question. That is "one dimensional" integral, in a single variable. How do you intend to change it to a three dimensional problem?
There is a method for integrating $\displaystyle \int_{-\infty}^\infty e^{-x^2}dx$ by: Squaring then writing it is the product of two integrals in x and y, then converting to polar coordinates. But that method won't work here because it depends upon the fact that [tex]e^{-x^2}[/math ] goes to 0 at both $\displaystyle \infty$ and $\displaystyle -\infty$.