# Primitive of Power of x by Logarithm of x

## Theorem

$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$

where $m \ne -1$.

## 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:

 $\ds u$ $=$ $\ds \ln x$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds \frac 1 x$ Derivative of $\ln x$

and let:

 $\ds \frac {\d v} {\d x}$ $=$ $\ds x^m$ $\ds \leadsto \ \$ $\ds v$ $=$ $\ds \frac {x^{m + 1} } {m + 1}$ Primitive of Power

Then:

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

$\blacksquare$

## Example

$\ds \int_1^n x^m \ln x \rd x = \frac {n^{m + 1} } {m + 1} \paren {\ln n - \frac 1 {m + 1} } + \frac 1 {\paren {m + 1}^2}$

where $m \ne -1$.