Primitive of Reciprocal of x cubed by x squared plus a squared squared/Partial Fraction Expansion
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Lemma for Primitive of Reciprocal of $x^3 \paren {x^2 + a^2}^2$
- $\dfrac 1 {x^3 \paren {x^2 + a^2}^2} \equiv -\dfrac 2 {a^6 x} + \dfrac 1 {a^4 x^3} + \dfrac {2 x} {a^6 \paren {x^2 + a^2} } + \dfrac x {a^4 \paren {x^2 + a^2}^2}$
Proof
| \(\ds \dfrac 1 {x^3 \paren {x^2 + a^2}^2}\) | \(\equiv\) | \(\ds \dfrac A x + \dfrac B {x^2} + \dfrac C {x^3} + \dfrac {D x + E} {x^2 + a^2} + \dfrac {F x + G} {\paren {x^2 + a^2}^2}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1\) | \(\equiv\) | \(\ds A x^2 \paren {x^2 + a^2}^2 + B x \paren {x^2 + a^2}^2 + C \paren {x^2 + a^2}^2\) | multiplying through by $x^2 \paren {x^2 + a^2}^2$ | ||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {D x + E} x^3 \paren {x^2 + a^2} + \paren {F x + G} x^3\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds \) | \(\equiv\) | \(\ds A x^6 + 2 A a^2 x^4 + A a^4 x^2 + B x^5 + 2 B a^2 x^3 + B a^4 x\) | multiplying everything out | ||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds C x^4 + 2 C a^2 x^2 + C a^4\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds D x^6 + D x^4 a^2 + E x^5 + E x^3 a^2 + F x^4 + G x^3\) |
Setting $x = 0$ in $(1)$:
| \(\ds C a^4\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds C\) | \(=\) | \(\ds \frac 1 {a^4}\) |
Equating coefficients of $x^2$ in $(1)$:
| \(\ds 0\) | \(=\) | \(\ds A a^4 + 2 C a^2\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds A\) | \(=\) | \(\ds -\frac 2 {a^6}\) |
Equating coefficients of $x^6$ in $(1)$:
| \(\ds 0\) | \(=\) | \(\ds A + D\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds D\) | \(=\) | \(\ds \frac 2 {a^6}\) |
Equating coefficients of $x$ in $(1)$:
| \(\ds B\) | \(=\) | \(\ds 0\) |
Equating coefficients of $x^5$ in $(1)$:
| \(\ds 0\) | \(=\) | \(\ds B + E\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds E\) | \(=\) | \(\ds 0\) |
Equating coefficients of $x^3$ in $(1)$:
| \(\ds 0\) | \(=\) | \(\ds 2 B a^2 + E a^2 + G\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds G\) | \(=\) | \(\ds 0\) |
Equating coefficients of $x^4$ in $(1)$:
| \(\ds 2 A a^2 + C + D a^2 + F\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds F\) | \(=\) | \(\ds \frac 1 {a^4}\) |
Summarising:
| \(\ds A\) | \(=\) | \(\ds -\frac 2 {a^6}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds C\) | \(=\) | \(\ds \frac 1 {a^4}\) | ||||||||||||
| \(\ds D\) | \(=\) | \(\ds \frac 2 {a^6}\) | ||||||||||||
| \(\ds E\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds F\) | \(=\) | \(\ds \frac 1 {a^4}\) | ||||||||||||
| \(\ds G\) | \(=\) | \(\ds 0\) |
Hence the result.
$\blacksquare$