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

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Theorem

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


Proof

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

$\blacksquare$


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Sources