求定积分∫（ x^3)[e^(-x^2)] dx 上限(ln2)^1/2,下限0

题目

求定积分∫（ x^3)[e^(-x^2)] dx 上限(ln2)^1/2,下限0

答案

∫x³e^(-x²) dx
=-1/2∫x²de^(-x²)
=-1/2x²e^(-x²)上限(ln2)^1/2,下限0+∫e^(-x²)xdx
=-1/2x²e^(-x²)上限(ln2)^1/2+∫e^(-x²)xdx
=-1/4ln2+∫e^(-x²)xdx
=-1/4ln2-1/2∫de^(-x²)
=-1/4ln2-1/2e^(-x²)上限(ln2)^1/2,下限0
=-1/4ln2+1/4