\section{Primitive of Power of x by Power of Logarithm of x}
Tags: Primitives involving Logarithm Function

\begin{theorem}
:$\ds \int x^m \ln^n x \rd x = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \rd x + C$
where $m \ne -1$.
\end{theorem}

\begin{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:
{{begin-eqn}}
{{eqn | l = u
      | r = \ln^n x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = n \ln^{n - 1} x \frac 1 x
      | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = x^m
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {x^{m + 1} } {m + 1}
      | c = Primitive of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \ln^n x \rd x
      | r = \frac {x^{m + 1} } {m + 1} \ln^n x - \int \frac {x^{m + 1} } {m + 1} \paren {n \ln^{n - 1} x \frac 1 x} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \rd x + C
      | c = Primitive of Constant Multiple of Function
}}
{{end-eqn}}
{{qed}}
\end{proof}

