Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of x
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Theorem
- $\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$
Proof 1
From Reduction Formula for Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
- $\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Substituting $m + 1$ for $m$:
\(\ds \int x^{m + 1} \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + n + 2} a} - \frac {\paren {m + 1} b} {\paren {m + n + 2} a} \int x^m \paren {a x + b}^n \rd x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\paren {m + 1} b} {\paren {m + n + 2} a} \int x^m \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + n + 2} a} - \int x^{m + 1} \paren {a x + b}^n \rd x\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int x^m \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + n + 2} a} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x\) | rearranging | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x\) | rearranging |
$\blacksquare$
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 $p := 1, q := 0, m := n, n := m$:
\(\ds \int x^m \paren {a x + b}^n \rd x\) | \(=\) | \(\ds \frac 1 {\paren {m + 1} \paren {b 1 - a 0} } \paren {\paren {a x + b}^{n + 1} \paren {1 x + 0}^{m + 1} - a \paren {m + n + 2} \int \paren {a x + b}^n \paren {0 x + 1}^{m + 1} \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\paren {m + 1} b} \paren {\paren {a x + b}^{n + 1} x^{m + 1} - a \paren {m + n + 2} \int \paren {a x + b}^n x^{m + 1} \rd x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x\) |
$\blacksquare$