Question

已知f(x)的导函数为e的x次方/x,求f(x)的一个原函数

Answer 1

∫ f(x)dx = (1/x)e^x

f(x) = (xe^x-e^x)/x² = (1/x²)(x-1)e^x

∫ xf'(x) dx

= ∫ x df(x)

= xf(x) - ∫ f(x)dx

= (1/x)(x-1)e^x - (1/x)e^x + C

= (1/x)(x-2)e^x + C