What is f(x) = int 1/x  if f(2)=1 ?

May 3, 2016

$\ln \left(\frac{x}{2}\right) + 1$

Explanation:

The derivative of $\ln x = \frac{1}{x}$

hence the anti-derivative of $\frac{1}{x} \text{ is} \ln x$

$\Rightarrow F \left(x\right) = \int \frac{1}{x} \mathrm{dx} = \ln x + c$

To find c , use f(2) = 1

ln2 + c = 1 → c = 1 - ln2

$\Rightarrow F \left(x\right) = \ln x + 1 - \ln 2$

using • lnx-lny=ln(x/y)"to simplify"

$\Rightarrow \int \frac{1}{x} \mathrm{dx} = \ln \left(\frac{x}{2}\right) + 1$