Primitives involving x squared minus a squared squared

Theorem

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

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

$\ds \int \frac {\d x} {\paren {x^2 - a^2}^2} = \frac {-x} {2 a^2 \paren {x^2 - a^2} } + \frac 1 {4 a^3} \map \ln {\frac {x + a} {x - a} } + C$

for $x^2 > a^2$.

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

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

for $x^2 > a^2$.

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

$\ds \int \frac {x^2 \rd x} {\paren {x^2 - a^2}^2} = \frac {-x} {2 \paren {x^2 - a^2} } + \frac 1 {4 a} \map \ln {\frac {x - a} {x + a} } + C$

for $x^2 > a^2$.

Primitive $x^3$ over $\paren {x^2 - a^2}^2$

$\ds \int \frac {x^3 \rd x} {\paren {x^2 - a^2}^2} = \frac {-a^2} {2 \paren {x^2 - a^2} } + \frac 1 2 \map \ln {x^2 - a^2} + C$

for $x^2 > a^2$.

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

$\ds \int \frac {\d x} {x \paren {x^2 - a^2}^2} = \frac {-1} {2 a^2 \left({x^2 - a^2}\right)} + \frac 1 {2 a^4} \ln \left({\frac {x^2} {x^2 - a^2} }\right) + C$

for $x^2 > a^2$.

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

$\ds \int \frac {\d x} {x^2 \paren {x^2 - a^2}^2} = \frac {-1} {a^4 x} - \frac x {2 a^4 \paren {x^2 - a^2} } + \frac 3 {4 a^5} \map \ln {\frac {x + a} {x - a} } + C$

for $x^2 > a^2$.

Primitive of Reciprocal of $x^3 \paren {x^2 - a^2}^2$

$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2} = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 - a^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 - a^2} } + C$

for $x^2 > a^2$.

Also see

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