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

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

where $m \ne -1$.


Proof

\(\ds \int_1^n x^m \ln x \rd x\) \(=\) \(\ds \intlimits {\frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } } 1 n\) Primitive of Power of x by Logarithm of x
\(\ds \) \(=\) \(\ds \paren {\frac {n^{m + 1} } {m + 1} \paren {\ln n - \frac 1 {m + 1} } } - \paren {\frac {1^{m + 1} } {m + 1} \paren {\ln 1 - \frac 1 {m + 1} } }\) Definition of Definite Integral
\(\ds \) \(=\) \(\ds \frac {n^{m + 1} } {m + 1} \paren {\ln n - \frac 1 {m + 1} } - \frac 1 {m + 1} \paren {-\frac 1 {m + 1} }\) Logarithm of 1 is 0
\(\ds \) \(=\) \(\ds \frac {n^{m + 1} } {m + 1} \paren {\ln n - \frac 1 {m + 1} } + \frac 1 {\paren {m + 1}^2}\) Logarithm of 1 is 0

$\blacksquare$


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