设f(x)=1/(1+x²)+e^x∫(0积到1)f(x)dx，试求:∫(0积到1)f(x)dx.

Question

如题所述

Answer 1

∫[1+x
f'(x)]e^
f(x)dx
=∫e^
f(x)dx+∫xf'(x)e^
f(x)dx
‘先将括号打开，拆为两个积分
=∫e^
f(x)dx+∫xe^
f(x)df(x)
=∫e^
f(x)dx+∫xde^
f(x)
=∫e^
f(x)dx+xe^
f(x)-∫e^
f(x)dx
'分部积分法则
=xe^
f(x)
最后结果1*e^
f(1)-0*e^
f(0)=e^
f(1)