Primitive of Reciprocal of Power of x by Power of x squared plus a squared
Jump to navigation
Jump to search
Theorem
- $\ds \int \frac {\d x} {x^m \paren {x^2 + a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} } - \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 + a^2}^n}$
Proof
| \(\ds \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} }\) | \(=\) | \(\ds \int \frac {\paren {x^2 + a^2} \rd x} {x^m \paren {x^2 + a^2}^{n - 1} \paren {x^2 + a^2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {\paren {x^2 + a^2} \rd x} {x^m \paren {x^2 + a^2}^{\left({n - 1}\right) + 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac {x^2 \rd x} {x^m \paren {x^2 + a^2}^n} + a^2 \int \frac {\d x} {x^m \paren {x^2 + a^2}^n}\) | 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 | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a^2 \int \frac {\d x} {x^m \paren {x^2 + a^2}^n}\) | \(=\) | \(\ds \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} } - \int \frac {\d x} {x^{m - 2} \paren {x^2 + a^2}^n}\) | changing sides | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {x^m \paren {x^2 + a^2}^n}\) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} } - \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 + a^2}^n}\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^2 + a^2$: $14.143$