Primitive of Power of x by Logarithm of x
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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:
\(\ds \int x^m \ln x \rd x\) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \ln x - \int \frac {x^{m + 1} } {m + 1} \paren {\frac 1 x} \rd x\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^{m + 1} } {m + 1} \ln x - \frac 1 {m + 1} \int x^m \rd x\) | 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 $m \ne -1$.
Also see
- Primitive of $\dfrac {\ln x} x$ for the case where $m = -1$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\ln x$: $14.527$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals