Primitive of Reciprocal of Power of x by Power of a squared minus x squared

Theorem

$\ds \int \frac {\d x} {x^m \paren {a^2 - x^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^m \paren {a^2 - x^2}^{n - 1} } + \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {a^2 - x^2}^n}$

for $x^2 < a^2$.

Proof

\(\ds \int \frac {\d x} {x^m \paren {a^2 - x^2}^{n - 1} }\)

\(=\)

\(\ds \int \frac {\paren {a^2 - x^2} \rd x} {x^m \paren {a^2 - x^2}^{n - 1} \paren {a^2 - x^2} }\)

\(\ds \)

\(=\)

\(\ds \int \frac {\paren {a^2 - x^2} \rd x} {x^m \paren {a^2 - x^2}^{\paren {n - 1} + 1} }\)

\(\ds \)

\(=\)

\(\ds a^2 \int \frac {\d x} {x^m \paren {a^2 - x^2}^n} - \int \frac {x^2 \rd x} {x^m \paren {a^2 - x^2}^n}\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds a^2 \int \frac {\d x} {x^m \paren {a^2 - x^2}^n} - \int \frac {\d x} {x^{m - 2} \paren {a^2 - x^2}^n}\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds a^2 \int \frac {\d x} {x^m \paren {a^2 - x^2}^n}\)

\(=\)

\(\ds \int \frac {\d x} {x^m \paren {a^2 - x^2}^{n - 1} } + \int \frac {\d x} {x^{m - 2} \paren {a^2 - x^2}^n}\)

changing sides

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d x} {x^m \paren {a^2 - x^2}^n}\)

\(=\)

\(\ds \frac 1 {a^2} \int \frac {\d x} {x^m \paren {a^2 - x^2}^{n - 1} } + \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {a^2 - x^2}^n}\)

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {x^m \paren {x^2 - a^2}^n}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.181$