Primitive of Power of Logarithm of x

Theorem

$\ds \int \ln^n x \rd x = x \ln^n x - n \int \ln^{n - 1} x \rd x + C$

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 \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds n \ln^{n - 1} x \frac 1 x$

and let:

$\ds \frac {\d v} {\d x}$

$=$

$\ds 1$

$\ds \leadsto \ \ $

$\ds v$

$=$

$\ds x$

Then:

$\ds \int \ln^n x \rd x$

$=$

$\ds x \ln^n x - \int x \paren {n \ln^{n - 1} x \frac 1 x} \rd x + C$

$\ds $

$=$

$\ds x \ln^n x - n \int \ln^{n - 1} x \rd x + C$

$\blacksquare$

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