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

## 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

$\blacksquare$