Primitives involving Power of a squared minus x squared

Theorem

This page gathers together the primitives of some expressions involving $\paren {x^2 - a^2}^n$.

Primitive of Reciprocal of $\paren {x^2 - a^2}^n$

$\ds \int \frac {\d x} {\paren {a^2 - x^2}^n} = \frac x {2 \paren {n - 1} a^2 \paren {a^2 - x^2}^{n - 1} } + \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {a^2 - x^2}^{n - 1} }$

for $x^2 > a^2$.

Primitive $x$ over $\paren {x^2 - a^2}^n$

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

for $x^2 < a^2$.

Primitive of Reciprocal of $x \paren {x^2 - a^2}^n$

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

for $x^2 < a^2$.

Primitive of $x^m$ over $\paren {x^2 - a^2}^2$

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

for $x^2 < a^2$.

Primitive of Reciprocal of $x^m \paren {x^2 - a^2}^n$

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

for $x^2 < a^2$.

Also see

• Primitives involving $a^2 - x^2$
• Primitives involving $\paren {x^2 - a^2}^2$