Primitive of Function of Natural Logarithm

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Theorem

$\displaystyle \int F \left({\ln x}\right) \rd x = \int F \left({u}\right) e^u \rd u$

where $u = \ln x$.


Proof

\(\displaystyle u\) \(=\) \(\displaystyle \ln x\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle \frac 1 x\) Derivative of Natural Logarithm Function
\(\displaystyle \implies \ \ \) \(\displaystyle \int F \left({\ln x}\right) \rd x\) \(=\) \(\displaystyle \int F \left({u}\right) \ x \rd u\) Primitive of Composite Function
\(\displaystyle \) \(=\) \(\displaystyle \int F \left({u}\right) \ e^u \rd u\) Definition of Exponential Function

$\blacksquare$


Sources