Primitive of Reciprocal of x by x squared minus a squared squared
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Theorem
- $\ds \int \frac {\d x} {x \paren {x^2 - a^2}^2} = \frac {-1} {2 a^2 \left({x^2 - a^2}\right)} + \frac 1 {2 a^4} \ln \left({\frac {x^2} {x^2 - a^2} }\right) + C$
for $x^2 > a^2$.
Proof
\(\ds \int \frac {\d x} {x \paren {x^2 - a^2}^2}\) | \(=\) | \(\ds \int \left({\frac 1 {a^4 x} + \frac {-x} {a^4 \paren {x^2 - a^2} } + \frac x {a^2 \paren {x^2 - a^2}^2} }\right) \rd x\) | Partial Fraction Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^4} \int \frac {\d x} x + \frac {-1} {a^4} \int \frac {x \rd x} {x^2 - a^2} + \frac 1 {a^2} \int \frac {x \rd x} {\paren {x^2 - a^2}^2}\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^4} \ln \size x + \frac {-1} {a^4} \int \frac {x \rd x} {x^2 - a^2} + \frac 1 {a^2} \int \frac {x \rd x} {\paren {x^2 - a^2}^2} + C\) | Primitive of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^4} \ln \size x + \frac {-1} {a^4} \paren {\frac 1 2 \map \ln {x^2 - a^2} } + \frac 1 {a^2} \int \frac {x \rd x} {\paren {x^2 - a^2}^2} + C\) | Primitive of $\dfrac x {x^2 - a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {a^4} \ln \size x + \frac {-1} {2 a^4} \map \ln {x^2 - a^2} + \frac 1 {a^2} \paren {\frac {-1} {2 \paren {x^2 - a^2} } } + C\) | Primitive of $\dfrac x {\paren {x^2 - a^2}^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a^4} \map \ln {x^2} + \frac {-1} {2 a^4} \map \ln {x^2 - a^2} + \frac 1 {a^2} \paren {\frac {-1} {2 \paren {x^2 - a^2} } } + C\) | Logarithm of Power and $x^2 > 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-1} {2 a^2 \paren {x^2 - a^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {x^2 - a^2} } + C\) | Difference of Logarithms |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.155$