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

$\blacksquare$


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