Primitive of Reciprocal of x cubed by Root of a squared minus x squared cubed

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Theorem

$\ds \int \frac {\d x} {x^3 \paren {\sqrt {a^2 - x^2} }^3} = \frac {-1} {2 a^2 x^2 \sqrt {a^2 - x^2} } + \frac 3 {2 a^4 \sqrt {a^2 - x^2} } - \frac 3 {2 a^5} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$


Proof

\(\ds \int \frac {\d x} {x^3 \paren {\sqrt {a^2 - x^2} }^3}\) \(=\) \(\ds \int \frac {a^2 \rd x} {a^2 x^3 \paren {\sqrt {a^2 - x^2} }^3}\)
\(\ds \) \(=\) \(\ds \int \frac {\paren {a^2 - x^2 + x^2} \rd x} {a^2 x^3 \paren {\sqrt {a^2 - x^2} }^3}\)
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\paren {a^2 - x^2} \rd x} {x^3 \paren {\sqrt {a^2 - x^2} }^3} + \frac 1 {a^2} \int \frac {x^2 \rd x} {x^3 \paren {\sqrt {a^2 - x^2} }^3}\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} } + \frac 1 {a^2} \int \frac {\d x} {x \paren {\sqrt {a^2 - x^2} }^3}\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {\frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } } + \frac 1 {a^2} \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3} + C\) Primitive of $\dfrac 1 {x^3 \sqrt {a^2 - x^2} }$
\(\ds \) \(=\) \(\ds \frac 1 {a^2} \paren {\frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \frac 1 {a^2} \paren {\frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } } + C\) Primitive of $\dfrac 1 {x \paren {\sqrt {a^2 - x^2} }^3}$
\(\ds \) \(=\) \(\ds \frac {-1} {2 a^2 x^2 \sqrt {a^2 - x^2} } + \frac 3 {2 a^4 \sqrt {a^2 - x^2} } - \frac 3 {2 a^5} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C\) simplification

$\blacksquare$


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