Primitive of Reciprocal of x by Logarithm of x

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Theorem

$\ds \int \frac {\d x} {x \ln x} = \ln \size {\ln x} + C$


Proof

\(\ds z\) \(=\) \(\ds \ln x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds \frac 1 x\) Derivative of Natural Logarithm
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {x \ln x}\) \(=\) \(\ds \ln \size {\ln x} + C\) Primitive of Function under its Derivative

$\blacksquare$


Sources