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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