The difference an xx can make

Calculus Level 2

The Gaussian function is the function f(x)=ex2.f(x)=e^{-x^2}. While its antiderivative cannot be written with elementary functions, its definite integral can still be calculated. ex2 dx=π\int_{-\infty}^\infty e^{-x^2}\text{ d}x=\sqrt{\pi} Because of its properties as an even function, we can say this. 0ex2 dx=π2\int_0^\infty e^{-x^2}\text{ d}x=\dfrac{\sqrt{\pi}}{2} But a simple xx can make a big difference. Not only does g(x)=xex2g(x)=xe^{-x^2} have a simple antiderivative, but the area under it from 00 to \infty is a rational number! 0xex2 dx=AB\int_0^\infty xe^{-x^2}\text{ d}x=\dfrac{A}{B} If AA and BB are positive coprime integers, what is A+B?A+B\text{?}

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