求定积分∫【5/0】{x^3/（x^2+1)}dx

题目

求定积分∫【5/0】{x^3/（x^2+1)}dx

答案

∫{x^3/（x^2+1)}dx
=(1/2)∫x^2/(x^2+1)d(x^2)
=(1/2)∫[1-1/(x^2+1)]d(x^2)
=(1/2)∫d(x^2)-(1/2)∫1/(x^2+1)d(x^2)
=x^2/2-(1/2)ln(1+x^2)+C
上限5、下限0代入：
原式=12.5-0.5ln26