Primitives involving x squared minus a squared

Theorem

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

Primitive of Reciprocal of $x^2 - a^2$: $\coth^{-1}$ form

Let $a \in \R_{>0}$ be a strictly positive real constant.

Let $\size x > a$.

Then:

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

Primitive of Reciprocal of $x^2 - a^2$: Logarithm Form

Let $a \in \R_{>0}$ be a strictly positive real constant.

Let $x \in \R$ such that $\size x \ne a$.

$1$st Logarithm Form

$\ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a\\

& \\ \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\ & \\ \text {undefined} & : \size x = a \end {cases}$

$2$nd Logarithm Form

$\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$

Primitive $x$ over $x^2 - a^2$

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

for $x^2 > a^2$.

Theorem

Logarithm Form

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

for $x^2 > a^2$.

Inverse Hyperbolic Cotangent Form

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

for $x^2 > a^2$.

Also see

• Primitive of $\dfrac {x^2} {a^2 - x^2}$

Primitive $x^3$ over $x^2 - a^2$

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

for $x^2 > a^2$.

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

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

for $x^2 > a^2$.

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

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

for $x^2 > a^2$.

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

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

for $x^2 > a^2$.

Also see

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