Question

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

Answer 1

`f'(x)=\frac2x`

`\implies f(x)=\displaystyle\frac2x\,\mathrm dx=2\ln|x|+C`


`f(\sqrt e)=5\implies 5=2\ln|\sqrt e|+C`

`\implies 5=\frac22\ln e+C`

`\implies C=4`

`\implies f(x)=2\ln|x|+4`

`\implies f(e)=2\ln|e|+4=6`