Primitive of General Logarithm Function
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Theorem
| \(\ds \int \log_a x \rd x\) | \(=\) | \(\ds \dfrac 1 {\ln a} \paren {x \ln x - x} + C\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x \log_a x - \dfrac x {\ln a}\) |
Proof
| \(\ds \int \log_a x \rd x\) | \(=\) | \(\ds \int \dfrac {\ln x} {\ln a} \rd x\) | Change of Base of Logarithm | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {\ln a} \paren {x \ln x - x} + C\) | Primitive of $\ln x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {x \ln x} {\ln a} \dfrac x {\ln a} + C\) | rearranging | |||||||||||
| \(\ds \) | \(=\) | \(\ds x \log_a x - \dfrac x {\ln a} + C\) | Change of Base of Logarithm |
$\blacksquare$
Sources
- 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Appendix $\text I$: Table of Indefinite Integrals $11$.