Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1

Theorem

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

Let $x \in \R$.

Then:

$\ds \int \frac {\d x} {a^2 - x^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}$

Proof

Case where $\size x < a$

Let $\size x < a$.

Then:

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

Case where $\size x > a$

Let $\size x > a$.

Then:

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

Also see

• Primitive of $\dfrac 1 {x^2 + a^2}$
• Primitive of $\dfrac 1 {x^2 - a^2}$: Logarithm Form