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

From ProofWiki
Jump to navigation Jump to search

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

$\blacksquare$


Also see


Sources