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

## Theorem

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

## Proof

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

$\blacksquare$