Primitive of Power of a x + b
Theorem
- $\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
Proof 1
Let $u = a x + b$.
Then:
| \(\ds \int \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac 1 a \int u^n \rd u\) | Primitive of Function of $a x + b$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 a \frac {u^{n + 1} } {n + 1} + C\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C\) | substituting for $u$ |
$\blacksquare$
Proof 2
Let $u = a x + b$.
Then:
- $\dfrac {\d u} {\d x} = a$
Then:
| \(\ds \int \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \int \dfrac {u^n} a \rd u\) | Integration by Substitution | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 a \dfrac {u^{n + 1} } {n + 1}\) | Primitive of Power | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C\) | substituting back for $u$ |
$\blacksquare$
Proof 3
| \(\ds \map {\dfrac \d {\d x} } {\frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} }\) | \(=\) | \(\ds \dfrac {\paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a} \map {\dfrac \d {\d x} } {a x + b}\) | Power Rule for Derivatives, Chain Rule for Derivatives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {a \paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a}\) | Power Rule for Derivatives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a x + b}^n\) | simplifying |
The result follows by definition of primitive.
$\blacksquare$
Also see
- Primitive of Reciprocal of $a x + b$ for the case when $n = -1$
Sources
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(i) (b)}$
- 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: $3.3.14$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$: $14.80$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Front endpapers: A Brief Table of Integrals: $5$.
- 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 $a x + b$: $17.1.14.$