Question

If f'(x)=2/x and f(sqrt(e))=5, then f(e)=

Answer 1

\[
dy={2\over x}dx\implies y=2\ln(x)+C=f(x)\\
\]
to get "C", use the condition
\[f(\sqrt{e})=2\ln(\sqrt{e})+C=5\\\implies{\cancel{2}\over\cancel{2}}\ln(e)+C=5\implies
1+C=5\implies C=4\\
\therefore f(x)=2\ln(x)+4
\]
now substitute x=e and evaluate!

Answer 2

Got it.