Primitive of Reciprocal of a x + b

Theorem

$\ds \int \frac {\d x} {a x + b} = \frac 1 a \ln \size {a x + b} + C$

where $a \ne 0$ and $x \ne - \dfrac b a$.

$\ds \int \frac {\d x} {a x + b}$

$=$

$\ds \frac 1 a \int \frac {\map \d {a x + b} } {a x + b}$

$\ds $

$=$

$\ds \frac 1 a \ln \size {a x + b} + C$

$\blacksquare$

$\ds \int \frac {\d x} {x - a} = \ln \size {x - a} + C$

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $5$.
1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(ii) (b)}$
1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integrals of Rational Algebraic Functions: $3.3.15$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$: $14.59$
1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $6$.
1983: K.G. Binmore: Calculus ... (previous) ... (next): $9$ Sums and Integrals: $9.8$ Standard Integrals
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.1.$