Primitive of Logarithm of x over x/Corollary

Corollary to Primitive of $\dfrac {\ln x} x$

$\ds \int \frac {\ln a x} x \rd x = \frac {\map {\ln^2} {a x} } 2 + C$

Proof

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: $z = a x$

\(\ds \)

\(=\)

\(\ds \int \frac {\ln z \rd z} z\)

\(\ds \)

\(=\)

\(\ds \frac {\ln^2 z} 2 + C\)

\(\ds \)

\(=\)

\(\ds \frac {\map {\ln^2} {a x} } 2 + C\)

$\blacksquare$