Question

设f'e^x)=x+1,求f(x)

Answer 1

f'(e^x)=x+1

e^x*f'(e^x)=(x+1)e^x

[f(e^x)]'=(x+1)e^x

两边积分得

f(e^x)=∫(x+1)e^xdx

=∫xe^xdx+∫e^xdx

=xe^x+C

令x=lnx代入得

f(x)=xlnx+C