Primitive of x cubed over Root of x squared minus a squared cubed

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Theorem

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

for $\size x > a$.


Proof

\(\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\) \(=\) \(\ds \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\)
\(\ds \) \(=\) \(\ds \int \frac {x \paren {x^2 - a^2 + a^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\)
\(\ds \) \(=\) \(\ds \int \frac {x \paren {x^2 - a^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3} + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \frac {x \rd x} {\sqrt {x^2 - a^2} } + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}\) simplifying
\(\ds \) \(=\) \(\ds \sqrt {x^2 - a^2} + a^2 \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3} + C\) Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$
\(\ds \) \(=\) \(\ds \sqrt {x^2 - a^2} + a^2 \frac {-1} {\sqrt {x^2 - a^2} } + C\) Primitive of $\dfrac x {\paren {\sqrt {x^2 - a^2} }^3}$
\(\ds \) \(=\) \(\ds \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C\) simplifying

$\blacksquare$


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Sources