Primitive of Reciprocal of a x + b

Theorem

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

Proof

\(\ds \int \frac {\d x} {a x + b}\)

\(=\)

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

Primitive of Function of $a x + b$

\(\ds \)

\(=\)

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

Primitive of Reciprocal

$\blacksquare$

Examples

Primitive of $\dfrac 1 {x - a}$

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

Sources

• 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: $\text {(ii) (b)}$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$: $14.59$
• 1983: K.G. Binmore: Calculus ... (previous) ... (next): $9$ Sums and Integrals: $9.8$ Standard Integrals