Primitive of Reciprocal of x squared by x cubed plus a cubed/Lemma

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Theorem

$\ds \int \frac {\d x} {x^2 \paren {x^3 + a^3} } = \frac {-1} {a^3 x} - \frac 1 {a^3} \int \frac {x \rd x} {\paren {x^3 + a^3} }$


Proof

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

$\blacksquare$