# Primitive of Function of Natural Logarithm

## 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

$\blacksquare$