# Primitive of Power of Logarithm of x

## Theorem

$\ds \int \ln^n x \rd x = x \ln^n x - n \int \ln^{n - 1} x \rd x + C$

## Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

 $\ds u$ $=$ $\ds \ln^n x$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds n \ln^{n - 1} x \frac 1 x$ Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives

and let:

 $\ds \frac {\d v} {\d x}$ $=$ $\ds 1$ $\ds \leadsto \ \$ $\ds v$ $=$ $\ds x$ Primitive of Constant

Then:

 $\ds \int \ln^n x \rd x$ $=$ $\ds x \ln^n x - \int x \paren {n \ln^{n - 1} x \frac 1 x} \rd x + C$ Integration by Parts $\ds$ $=$ $\ds x \ln^n x - n \int \ln^{n - 1} x \rd x + C$ Primitive of Constant Multiple of Function

$\blacksquare$