Primitive of Reciprocal of x cubed by x squared minus a squared squared

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Theorem

$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2} = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 - a^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 - a^2} } + C$

for $x^2 > a^2$.


Proof

\(\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2}\) \(=\) \(\ds \int \paren {\frac 1 {a^4 x^3} + \frac 2 {a^6 x} - \frac {2 x} {a^6 \paren {x^2 - a^2} } + \frac x {a^4 \paren {x^2 - a^2}^2} } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \frac 1 {a^4} \int \frac {\d x} {x^3} + \frac 2 {a^6} \int \frac {\d x} x - \frac 2 {a^6} \int \frac {x \rd x} {x^2 - a^2} + \frac 1 {a^4} \int \frac {x \rd x} {\paren {x^2 - a^2}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 {a^4} \frac {-1} {2 x^2} + \frac 2 {a^6} \int \frac {\d x} x - \frac 2 {a^6} \int \frac {x \rd x} {x^2 - a^2} + \frac 1 {a^4} \int \frac {x \rd x} {\paren {x^2 - a^2}^2} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-1} {2 a^4 x^2} + \frac 2 {a^6} \ln \size x - \frac 2 {a^6} \int \frac {x \rd x} {x^2 - a^2} + \frac 1 {a^4} \int \frac {x \rd x} {\paren {x^2 - a^2}^2} + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac {-1} {2 a^4 x^2} + \frac 2 {a^6} \ln \size x - \frac 2 {a^6} \frac 1 2 \map \ln {x^2 - a^2} + \frac 1 {a^4} \int \frac {x \rd x} {\paren {x^2 - a^2}^2} + C\) Primitive of $\dfrac x {x^2 - a^2}$
\(\ds \) \(=\) \(\ds \frac {-1} {2 a^4 x^2} + \frac 2 {a^6} \ln \size x - \frac 1 {a^6} \map \ln {x^2 - a^2} + \frac 1 {a^4} \frac {-1} {2 \paren {x^2 - a^2} } + C\) Primitive of $\dfrac x {\paren {x^2 - a^2}^2}$
\(\ds \) \(=\) \(\ds \frac {-1} {2 a^4 x^2} + \frac 1 {a^6} \map \ln {x^2} - \frac 1 {a^6} \map \ln {x^2 - a^2} + \frac 1 {a^4} \frac {-1} {2 \paren {x^2 - a^2} } + C\) Logarithm of Power and $x^2 > 0$
\(\ds \) \(=\) \(\ds \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 - a^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 - a^2} } + C\) Difference of Logarithms

$\blacksquare$


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