# Primitive of Logarithm of x over x/Corollary

## Corollary to Primitive of $\dfrac {\ln x} x$
$\ds \int \frac {\ln a x} x \rd x = \frac {\map {\ln^2} {a x} } 2 + C$
Let $z = a x$.
 $\ds z$ $=$ $\ds a x$ $\ds \leadsto \ \$ $\ds \d z$ $=$ $\ds a \rd x$ $\ds \leadsto \ \$ $\ds \int \frac {\ln a x} x \rd x$ $=$ $\ds \int \frac {\ln z} {z / a} \dfrac {\rd z} a$ Integration by Substitution: $z = a x$ $\ds$ $=$ $\ds \int \frac {\ln z \rd z} z$ $\ds$ $=$ $\ds \frac {\ln^2 z} 2 + C$ $\ds$ $=$ $\ds \frac {\map {\ln^2} {a x} } 2 + C$
$\blacksquare$