Primitive of x over x cubed plus a cubed

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Theorem

$\ds \int \frac {x \rd x} {x^3 + a^3} = \frac 1 {6 a} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {a \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$


Proof

\(\ds \int \frac {x \rd x} {x^3 + a^3}\) \(=\) \(\ds \int \frac {\paren {x + a - a} \rd x} {x^3 + a^3}\)
\(\ds \) \(=\) \(\ds \int \frac {\paren {x + a} \rd x} {x^3 + a^3} - a \int \frac {\d x} {x^3 + a^3}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {\paren {x + a} \rd x} {\paren {x + a} \paren {x^2 - a x + a^2} } - a \int \frac {\d x} {x^3 + a^3}\) Sum of Two Cubes
\(\ds \) \(=\) \(\ds \int \frac {\d x} {x^2 - a x + a^2} - a \int \frac {\d x} {x^3 + a^3}\) simplifying
\(\ds \) \(=\) \(\ds \frac 2 {a \sqrt 3} \arctan \paren {\frac {2 x - a} {a \sqrt 3} } - a \int \frac {\d x} {x^3 + a^3}\) Primitive of $\dfrac 1 {x^2 - a x + a^2}$
\(\ds \) \(=\) \(\ds \frac 2 {a \sqrt 3} \, \map \arctan {\frac {2 x - a} {a \sqrt 3} } - a \paren {\frac 1 {6 a^2} \ln \size {\frac {\paren {x + a}^2} {x^2 - a x + a^2} } + \frac 1 {a^2 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3} }\) Primitive of $\dfrac 1 {x^3 + a^3}$
\(\ds \) \(=\) \(\ds \frac 2 {a \sqrt 3} \, \map \arctan {\frac {2 x - a} {a \sqrt 3} } - \frac 1 {6 a} \ln \size {\frac {\paren {x + a}^2} {x^2 - a x + a^2} } - \frac 1 {a \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {6 a} \, \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {a \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}\) gathering terms

$\blacksquare$


Sources

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