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 |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers