Primitive of Reciprocal of x by x squared minus a squared
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Theorem
- $\ds \int \frac {\d x} {x \paren {x^2 - a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2 - a^2} {x^2} } + C$
for $x^2 > a^2$.
Proof
\(\ds \int \frac {\d x} {x \paren {x^2 - a^2} }\) | \(=\) | \(\ds \int \paren {\frac x {a^2 \paren {x^2 - a^2} } - \frac 1 {a^2 x} } \rd x\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {x \rd x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \int \frac {x \rd x} {x^2 - a^2} - \frac 1 {a^2} \ln \size x + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^2} \paren {\frac 1 2 \map \ln {x^2 - a^2} } - \frac 1 {a^2} \ln \size x + C\) | Primitive of $\dfrac x {x^2 - a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a^2} \map \ln {x^2 - a^2} - \frac 1 {2 a^2} \ln \size {x^2} + C\) | Logarithm of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a^2} \map \ln {x^2 - a^2} - \frac 1 {2 a^2} \map \ln {x^2} + C\) | as $x^2 > 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a^2} \map \ln {\frac {x^2 - a^2} {x^2} } + C\) | Difference of Logarithms |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.148$