Primitives involving Power of x squared minus a squared
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Theorem
This page gathers together the primitives of some expressions involving $\left({x^2 - a^2}\right)^n$.
Primitive of Reciprocal of $\left({x^2 - a^2}\right)^n$
- $\ds \int \frac {\d x} {\paren {x^2 - a^2}^n} = \frac {-x} {2 \paren {n - 1} a^2 \paren {x^2 - a^2}^{n - 1} } - \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$.
Primitive $x$ over $\left({x^2 - a^2}\right)^n$
- $\ds \int \frac {x \rd x} {\paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$.
Primitive of Reciprocal of $x \left({x^2 - a^2}\right)^n$
- $\ds \int \frac {\d x} {x \paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} a^2 \paren {x^2 - a^2}^{n - 1} } - \frac 1 {a^2} \int \frac {\d x} {x \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$.
Primitive of $x^m$ over $\left({x^2 - a^2}\right)^2$
- $\ds \int \frac {x^m \rd x} {\paren {x^2 - a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^{n - 1} } + a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^n}$
for $x^2 > a^2$.
Primitive of Reciprocal of $x^m \left({x^2 - a^2}\right)^n$
- $\ds \int \frac {\d x} {x^m \paren {x^2 - a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$.