Primitive of x over x cubed plus a cubed squared
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Theorem
- $\ds \int \frac {x \rd x} {\paren {x^3 + a^3}^2} = \frac {x^2} {3 a^3 \paren {x^3 + a^3} } + \frac 1 {18 a^4} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {3 a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$
Proof
First a lemma:
Lemma
- $\ds \int \frac {x \rd x} {\paren {x^3 + a^3}^2} = \frac {x^2} {3 a^3 \paren {x^3 + a^3} } + \frac 1 {3 a^3} \int \frac {x \rd x} {\paren {x^3 + a^3} }$
$\Box$
Then:
\(\ds \int \frac {x \rd x} {\paren {x^3 + a^3}^2}\) | \(=\) | \(\ds \frac {x^2} {3 a^3 \paren {x^3 + a^3} } + \frac 1 {3 a^3} \int \frac {x \rd x} {\paren {x^3 + a^3} }\) | from Lemma | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} {3 a^3 \paren {x^3 + a^3} } + \frac 1 {3 a^3} \paren {\frac 1 {6 a} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {a \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3} }\) | Primitive of $\dfrac x {\paren {x^3 + a^3} }$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^2} {3 a^3 \paren {x^3 + a^3} } + \frac 1 {18 a^4} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {3 a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $x^3 + a^3$: $14.305$