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 |
$\blacksquare$