# The difference an $$x$$ can make

Calculus Level 3

The Gaussian function is the function $$f(x)=e^{-x^2}.$$ While its antiderivative cannot be written with elementary functions, its definite integral can still be calculated. $\int_{-\infty}^\infty e^{-x^2}\text{ d}x=\sqrt{\pi}$ Because of its properties as an even function, we can say this. $\int_0^\infty e^{-x^2}\text{ d}x=\dfrac{\sqrt{\pi}}{2}$ But a simple $$x$$ can make a big difference. Not only does $$g(x)=xe^{-x^2}$$ have a simple antiderivative, but the area under it from $$0$$ to $$\infty$$ is a rational number! $\int_0^\infty xe^{-x^2}\text{ d}x=\dfrac{A}{B}$ If $$A$$ and $$B$$ are positive coprime integers, what is $$A+B\text{?}$$

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