# Primitive of Power of x by Logarithm of x/Example

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## Example of Primitive of Power of x by Logarithm of x

$\displaystyle \int_1^n x^m \ln x \rd x = \frac {n^{m + 1} } {m + 1} \left({\ln n - \frac 1 {m + 1} }\right) + \frac 1 {\left({m + 1}\right)^2}$

where $m \ne -1$.

## Proof

 $\displaystyle \int_1^n x^m \ln x \rd x$ $=$ $\displaystyle \left[{\frac {x^{m + 1} } {m + 1} \left({\ln x - \frac 1 {m + 1} }\right)}\right]_1^n$ Primitive of Power of x by Logarithm of x $\displaystyle$ $=$ $\displaystyle \left({\frac {n^{m + 1} } {m + 1} \left({\ln n - \frac 1 {m + 1} }\right)}\right) - \left({\frac {1^{m + 1} } {m + 1} \left({\ln 1 - \frac 1 {m + 1} }\right)}\right)$ Definition of Definite Integral $\displaystyle$ $=$ $\displaystyle \frac {n^{m + 1} } {m + 1} \left({\ln n - \frac 1 {m + 1} }\right) - \frac 1 {m + 1} \left({-\frac 1 {m + 1} }\right)$ Logarithm of 1 is 0 $\displaystyle$ $=$ $\displaystyle \frac {n^{m + 1} } {m + 1} \left({\ln n - \frac 1 {m + 1} }\right) + \frac 1 {\left({m + 1}\right)^2}$ Logarithm of 1 is 0

$\blacksquare$