Primitive of Logarithm of x over x/Corollary

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

$\blacksquare$