求不定积分e^(x^1/3)dx

Question

求不定积分e ^(x^1/3)dx

Answer 1

设t=x^(1/3),x=t^3,
dx=3t^2dt,
原式=∫e^t*3t^2dt
=3(t^2e^t-2∫t*e^tdt)
=3[t^2*e^t-2(te^t-∫e^tdt)]
=3t^2*e^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