Primitive of Power of Logarithm of x
Jump to navigation
Jump to search
Theorem
- $\ds \int \ln^n x \rd x = x \ln^n x - n \int \ln^{n - 1} x \rd x + C$
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:
\(\ds u\) | \(=\) | \(\ds \ln^n x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds n \ln^{n - 1} x \frac 1 x\) | Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds x\) | Primitive of Constant |
Then:
\(\ds \int \ln^n x \rd x\) | \(=\) | \(\ds x \ln^n x - \int x \paren {n \ln^{n - 1} x \frac 1 x} \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds x \ln^n x - n \int \ln^{n - 1} x \rd x + C\) | Primitive of Constant Multiple of Function |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\ln x$: $14.535$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $3$: Reduction formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $3$: Reduction formulae
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals: Reduction Formulae