Primitive of Reciprocal of a squared minus x squared squared/Partial Fraction Expansion
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Lemma for Primitive of Reciprocal of $\paren {a^2 - x^2}^2$
- $\dfrac 1 {\paren {a^2 - x^2}^2} \equiv \dfrac 1 {4 a^2} \paren {\dfrac 1 {\paren {a - x}^2} + \dfrac 1 {\paren {a + x}^2} + \dfrac 1 {a \paren {a + x} } + \dfrac 1 {a \paren {a - x} } }$
Proof
| \(\ds \dfrac 1 {\paren {a^2 - x^2}^2}\) | \(=\) | \(\ds \dfrac 1 {\paren {a + x}^2 \paren {a - x}^2}\) | Difference of Two Squares | |||||||||||
| \(\ds \) | \(\equiv\) | \(\ds \dfrac A {a + x} + \dfrac B {\paren {a + x}^2} + \dfrac C {a - x} + \dfrac D {\paren {a - x}^2}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1\) | \(\equiv\) | \(\ds A \paren {a^2 - x^2} \paren {a - x} + B \paren {a - x}^2 + C \paren {a^2 - x^2} \paren {a + x} + D \paren {a + x}^2\) | multiplying through by $\paren {a^2 - x^2}^2$ | ||||||||||
| \(\text {(1)}: \quad\) | \(\ds \) | \(\equiv\) | \(\ds A a^3 - A a^2 x - A a x^2 + A x^3 + B a^2 - 2 B a x + B x^2\) | multiplying out | ||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds C a^3 + C a^2 x - C a x^2 - C x^3 + D a^2 + 2 D a x + D x^2\) |
Setting $x = a$ in $(1)$:
| \(\ds D \paren {2 a}^2\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds D\) | \(=\) | \(\ds \frac 1 {4 a^2}\) |
Setting $x = -a$ in $(1)$:
| \(\ds B \paren {2 a}^2\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds B\) | \(=\) | \(\ds \frac 1 {4 a^2}\) |
Equating coefficients of $x^3$ in $(1)$:
| \(\ds 0\) | \(=\) | \(\ds A - C\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds C\) |
Equating coefficients of $x^2$ in $(1)$:
| \(\ds - A a - C a + B + D\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds - A a - C a + \frac 1 {4 a^2} + \frac 1 {4 a^2}\) | \(=\) | \(\ds 0\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -C a - C a + \frac 1 {2 a^2}\) | \(=\) | \(\ds 0\) | as $A = C$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 C\) | \(=\) | \(\ds \frac 1 {2 a^3}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds C\) | \(=\) | \(\ds \frac 1 {4 a^3}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds \frac 1 {4 a^3}\) |
Summarising:
| \(\ds A\) | \(=\) | \(\ds \frac 1 {4 a^3}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \frac 1 {4 a^2}\) | ||||||||||||
| \(\ds C\) | \(=\) | \(\ds \frac 1 {4 a^3}\) | ||||||||||||
| \(\ds D\) | \(=\) | \(\ds \frac 1 {4 a^2}\) |
Hence the result.
$\blacksquare$