Question

F'(x)=1/x, F(-1)=1, F(1)=0. Atunci F(e)+F(-e) este

Answer 1

F'(x) = 1/x
[tex]F(x) = \int\limits \,\frac{1}{x} dx \Rightarrow F(x) = ln|x| + C \\ \\ F(-1)=1 \Rightarrow ln|-1| + C = 1 \Rightarrow 0+C = 1 \Rightarrow C = 1 \\ F(1) = 0 \Rightarrow ln|1| + C = 0 \Rightarrow C = 0[/tex]

$F(x) = \left \{ {{ln(x)+1 ,x\ \le \ 0} \atop {ln(x),x\ \textgreater \ 0}} \right.$

Cand x  pozitiv, C = 0, iar cand x negativ C = 1.

$F(e) + F(-e) = ln|e|+ ln|-e| + 1 = 1+1+1 = 3$