Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2
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Theorem
- $\ds \int \frac {\d x} {x^2 - a^2} = \frac 1 {2 a} \ln \size {\frac {x - a} {x + a} } + C$
Proof
\(\ds \int \frac {\d x} {x^2 - a^2}\) | \(=\) | \(\ds \int \frac {\d x} {\paren {x - a} \paren {x + a} }\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \int \frac {\d x} {x - a} - \frac 1 {2 a} \int \frac {\d x} {x + a}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \ln \size {x - a} - \frac 1 {2 a} \ln \size {x + a} + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {2 a} \ln \size {\dfrac {x - a} {x + a} } + C\) | Difference of Logarithms |
$\blacksquare$