Primitive of Reciprocal of Power of x by Power of x squared minus a squared
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Theorem
- $\ds \int \frac {\d x} {x^m \paren {x^2 - a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$.
Proof
\(\ds \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }\) | \(=\) | \(\ds \int \frac {\paren {x^2 - a^2} \rd x} {x^m \paren {x^2 - a^2}^{n - 1} \paren {x^2 - a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {x^2 - a^2} \rd x} {x^m \paren {x^2 - a^2}^{\paren {n - 1} + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {x^2 \rd x} {x^m \paren {x^2 - a^2}^n} - a^2 \int \frac {\d x} {x^m \paren {x^2 - a^2}^n}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - a^2 \int \frac {\d x} {x^m \paren {x^2 - a^2}^n}\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a^2 \int \frac {\d x} {x^m \paren {x^2 - a^2}^n}\) | \(=\) | \(\ds \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }\) | changing sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {x^m \paren {x^2 - a^2}^n}\) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }\) |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.162$