Bonjour MinatoNM
[tex]1)\ F(x)=x\ln(x)-1\\\\F'(x)=[x\ln(x)]'-1'\\\\F'(x)=x'\times\ln(x)+x\times[\ln(x)]'-0\\\\F'(x)=1\times\ln(x)+x\times\dfrac{1}{x}\\\\\boxed{F'(x)=\ln(x)+1}[/tex]
[tex]2)\ I=\int\limits_1^e\ln x\ dx=\int\limits_1^e(\ln x+1-1)\ dx=\int\limits_1^e(\ln x+1)\ dx-\int\limits_1^e1 dx\\\\\\I=[F(x)]\limits_1^e-[x]\limits_1^e\\\\I=[x\ln(x)-1]\limits_1^e-[x]\limits_1^e\\\\I=[(e\ln(e)-1)-(1\ln(1)-1)]-[e-1]\\\\I=e-1-(0-1)-e+1\\\\\boxed{I=1}[/tex]
