Primitive of Reciprocal of x squared by x squared minus a squared

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Theorem

$\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^2 - a^2}\right)} = \frac 1 {a^2 x} + \frac 1 {2 a^3} \ln \left({\frac {x - a} {x + a} }\right) + C$

for $x^2 > a^2$.


Proof

\(\displaystyle \int \frac {\mathrm d x} {x^2 \left({x^2 - a^2}\right)}\) \(=\) \(\displaystyle \int \left({\frac 1 {a^2 \left({x^2 - a^2}\right)} - \frac 1 {a^2 x^2} }\right) \ \mathrm d x\) Partial Fraction Expansion
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \int \frac {\mathrm d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\mathrm d x} {x^2}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \int \frac {\mathrm d x} {x^2 - a^2} + \frac 1 {a^2 x} + C\) Primitive of Power
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \left({\frac 1 2 \ln \left ({\frac {x - a} {x + a} }\right) }\right) + \frac 1 {a^2 x} + C\) Primitive of $\dfrac 1 {x^2 - a^2}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2 x} + \frac 1 {2 a^3} \ln \left({\frac {x - a} {x + a} }\right) + C\) simplifying

$\blacksquare$


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