Primitive of Reciprocal of x by Logarithm of a x
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Theorem
- $\ds \int \frac {\d x} {x \ln a x} = \ln \size {\ln a x} + C$
Proof
| \(\ds z\) | \(=\) | \(\ds \ln a x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds \frac 1 x\) | Derivative of $\ln a x$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {x \ln a x}\) | \(=\) | \(\ds \ln \size {\ln a x} + C\) | Primitive of Function under its Derivative |
$\blacksquare$
Sources
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $112$.