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

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Theorem

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


Proof

\(\ds \int \frac {\d x} {x^3 \paren {x^4 - a^4} }\) \(=\) \(\ds \int \frac {a^4 \rd x} {a^4 x^3 \paren {x^4 - a^4} }\) multiplying top and bottom by $a^4$
\(\ds \) \(=\) \(\ds \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^3 \paren {x^4 - a^4} }\)
\(\ds \) \(=\) \(\ds \int \frac {\paren {x^4 - \paren {x^4 - a^4} } \rd x} {a^4 x^3 \paren {x^4 - a^4} }\)
\(\ds \) \(=\) \(\ds \frac {-1} {a^4} \int \frac {\paren {x^4 - a^4} \rd x} {x^3 \paren {x^4 - a^4} } + \frac 1 {a^4} \int \frac {x^4 \rd x} {x^3 \paren {x^4 - a^4} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {-1} {a^4} \int \frac {\d x} {x^3} + \frac 1 {a^4} \int \frac {x \rd x} {x^4 - a^4}\) simplifying
\(\ds \) \(=\) \(\ds \frac {-1} {a^4} \paren {\frac {-1} {2 x} } + \frac 1 {a^4} \int \frac {x \rd x} {x^4 - a^4}\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac 1 {2 a^4 x^2} + \frac 1 {a^4} \paren {\frac 1 {4 a^2} \ln \size {\frac {x^2 - a^2} {x^2 + a^2} } } + C\) Primitive of $\dfrac x {x^4 - a^4}$
\(\ds \) \(=\) \(\ds \frac 1 {2 a^4 x^2} + \frac 1 {4 a^6} \ln \size {\frac {x^2 - a^2} {x^2 + a^2} } + C\) simplifying

$\blacksquare$


Sources

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