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

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Theorem

$\displaystyle \int \frac {\mathrm d x} {x^3 \left({\sqrt {a^2 - x^2} }\right)^3} = \frac {-1} {2 a^2 x^2 \sqrt {a^2 - x^2} } + \frac 3 {2 a^4 \sqrt {a^2 - x^2} } - \frac 3 {2 a^5} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right) + C$


Proof

\(\displaystyle \int \frac {\mathrm d x} {x^3 \left({\sqrt {a^2 - x^2} }\right)^3}\) \(=\) \(\displaystyle \int \frac {a^2 \ \mathrm d x} {a^2 x^3 \left({\sqrt {a^2 - x^2} }\right)^3}\)
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\left({a^2 - x^2 + x^2}\right) \ \mathrm d x} {a^2 x^3 \left({\sqrt {a^2 - x^2} }\right)^3}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \int \frac {\left({a^2 - x^2}\right) \ \mathrm d x} {x^3 \left({\sqrt {a^2 - x^2} }\right)^3} + \frac 1 {a^2} \int \frac {x^2 \ \mathrm d x} {x^3 \left({\sqrt {a^2 - x^2} }\right)^3}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \int \frac {\mathrm d x} {x^3 \sqrt {a^2 - x^2} } + \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({\sqrt {a^2 - x^2} }\right)^3}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \left({\frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right)}\right) + \frac 1 {a^2} \int \frac {\mathrm d x} {x \left({\sqrt {x^2 - a^2} }\right)^3} + C\) Primitive of $\dfrac 1 {x^3 \sqrt {a^2 - x^2} }$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {a^2} \left({\frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right)}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {a^2} \left({\frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right)}\right) + C\) Primitive of $\dfrac 1 {x \left({\sqrt {a^2 - x^2} }\right)^3}$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-1} {2 a^2 x^2 \sqrt {a^2 - x^2} } + \frac 3 {2 a^4 \sqrt {a^2 - x^2} } - \frac 3 {2 a^5} \ln \left({\frac {a + \sqrt {a^2 - x^2} } x}\right) + C\) simplification

$\blacksquare$


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