∫e^x^1/3的不定积分?

题目

∫e^x^1/3的不定积分?

答案

被积函数是e^[x^(1/3)]?
令x^(1/3)=t
x=t^3
dx=3t^2dt
∫e^x^1/3dx
=∫e^t3t^2dt
=3∫t^2de^t
=3t^2e^t-6∫te^tdt
=3t^2e^t-6te^t+6∫e^tdt
=3t^2e^t-6te^t+6e^t+c
=3x^(2/3)e^(x^1/3)-6x^(1/3)e^(x^1/3)+6e^(x^1/3)+c