Answer
See proof
Work Step by Step
$$\eqalign{
& \int {\ln x} dx = x\ln x - x + C \cr
& {\text{Let }}F\left( x \right) = x\ln x - x + C{\text{ an antiderivative of }}\ln x,{\text{ we have to determine}} \cr
& \frac{d}{{dx}}\left[ {x\ln x - x + C} \right] \cr
& {\text{Differentiating }} \cr
& \underbrace {\frac{d}{{dx}}\left[ {x\ln x} \right]}_{{\text{Use product rule}}} - \frac{d}{{dx}}\left[ x \right] + \frac{d}{{dx}}\left[ C \right] \cr
& = x\frac{d}{{dx}}\left[ {\ln x} \right] + \ln x\frac{d}{{dx}}\left[ x \right] - \frac{d}{{dx}}\left[ x \right] + \frac{d}{{dx}}\left[ C \right] = \ln x \cr
& {\text{computing derivatives}} \cr
& = x\left( {\frac{1}{x}} \right) + \ln x\left( 1 \right) - 1 + 0 \cr
& {\text{Simplifying}} \cr
& 1 + \ln x - 1 = \ln x \cr
& {\text{The statement is true}}{\text{, then the formula is correct}}{\text{.}} \cr} $$
