Primitive of Function of a x + b

Theorem

$\ds \int \map F {a x + b} \rd x = \frac 1 a \int \map F u \rd u$

where $u = a x + b$.

$\ds u$

$=$

$\ds a x + b$

$\ds \leadsto \ \ $

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

$=$

$\ds a$

$\ds \leadsto \ \ $

$\ds \int \map F {a x + b} \rd x$

$=$

$\ds \int \frac {\map F u} a \d u$

$\ds $

$=$

$\ds \frac 1 a \int \map F u \rd u$

$\blacksquare$

1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Three rules for integration: $\text {III}$
1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Important Transformations: $14.49$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: Important Transformations: $16.49.$