Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of a x + b
Jump to navigation
Jump to search
Theorem
- $\ds \int x^m \paren {a x + b}^n \rd x = \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x$
Proof 1
From Reduction Formula for Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
- $\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Substituting $n + 1$ for $n$:
\(\ds \int x^m \paren {a x + b}^{n + 1} \rd x\) | \(=\) | \(\ds \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {m + n + 2} + \frac {\paren {n + 1} b} {m + n + 2} \int x^m \paren {a x + b}^n \rd x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\paren {n + 1} b} {m + n + 2} \int x^m \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {m + n + 2} + \int x^m \paren {a x + b}^{n + 1} \rd x\) | rearrangement | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int x^m \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac {m + n + 2} {\paren {n + 1} b} \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {m + n + 2} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x\) | rearrangement | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int x^m \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x\) | rearrangement |
$\blacksquare$
Proof 2
From Reduction Formula for Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power:
- $\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$
Setting $a := 1, b := 0, p x + q := a x + b$:
\(\ds \int x^m \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac 1 {\paren {n + 1} \paren {0 a - 1 b} } \paren {\paren {1 x + 0}^{m + 1} \paren {a x + b}^{n + 1} - 1 \paren {m + n + 2} \int \paren {1 x + 0}^m \paren {a x + b} ^{n + 1} \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\paren {n + 1} \paren {-b} } \paren {x^{m + 1} \paren {a x + b}^{n + 1} - \paren {m + n + 2} \int x^m \paren {a x + b}^{n + 1} \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x + b$: $14.83$