∫x^2*e^(x^2)dx和∫x^2*e^(-x^2)dx,不定積分均無法用初等函數(shù)表示,但∫x^2*e^(-x^2)dx在[0,+∞)上的定積分可求出
∫(0→+∞)x^2*e^(-x^2)dx
=∫(0→+∞)(-1/2)x*e^(x^2)d(-x^2)
=(-1/2)∫(0→+∞)x*d[e^(-x^2)]
=(-1/2){[x*e^(-x^2)]|(0→+∞) - ∫(0→+∞)e^(-x^2)dx}
=(1/2)∫(0→+∞)e^(-x^2)dx
=√π/4=(1/4)√π