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

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Theorem

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

where $x^2 > a^2$.


Proof 1

\(\displaystyle \forall x \in \R: x^2 > a^2: \ \ \) \(\displaystyle \int \frac {\d x} {x^2 - a^2}\) \(=\) \(\displaystyle -\frac 1 a \coth^{-1} {\frac x a} + C\) Primitive of Reciprocal of $x^2 - a^2$ in $\coth^{-1}$ form
\(\displaystyle \) \(=\) \(\displaystyle -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C\) $\coth^{-1} {\dfrac x a}$ in Logarithm Form
\(\displaystyle \) \(=\) \(\displaystyle -\frac 1 {2 a} \map \ln {\frac {x + a} {x - a} } + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 a} \map \ln {\frac {x - a} {x + a} } + C\) Logarithm of Reciprocal

$\blacksquare$


Proof 2

\(\displaystyle \int \frac {\d x} {x^2 - a^2}\) \(=\) \(\displaystyle \int \frac {\d x} {\paren {x - a} \paren {x + a} }\) Difference of Two Squares
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }\) Partial Fraction Expansion
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 a} \int \frac {\d x} {x - a} - \frac 1 {2 a} \int \frac {\d x} {x + a}\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 a} \ln \size {x - a} - \frac 1 {2 a} \ln \size {x + a} + C\) Primitive of Reciprocal
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {2 a} \ln \size {\dfrac {x - a} {x + a} } + C\) Difference of Logarithms

$\blacksquare$


Also see


Sources