Answer
$f(x)=-\ln|x|+(\ln 2)x-\ln 2$
Work Step by Step
$f''(x)=x^{-2}$
Using the antiderivatives table,
$f'(x)=\displaystyle \frac{x^{-1}}{-1}+C=-\frac{1}{x}+C$
Using the antiderivatives table,
$f(x)=-\ln|x|+Cx+D$
$\left\{\begin{array}{llll}
f(1)=0 & \Rightarrow & 0+C+D=0 & \\
& & D=-C & \\
& & & \\
f(2)=0 & \Rightarrow & & -\ln 2+2C-C=0\\
& & & C=\ln 2\\
& & & D=-\ln 2
\end{array}\right.$
$f(x)=-\ln|x|+(\ln 2)x-\ln 2$