Primitive of Root of x squared minus a squared over x cubed

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Theorem

$\displaystyle \int \frac {\sqrt {x^2 - a^2} } {x^3} \rd x = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 {2 a} \arcsec \size {\frac x a} + C$


Proof

Let:

\(\displaystyle z\) \(=\) \(\displaystyle x^2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d z} {\d x}\) \(=\) \(\displaystyle 2 x\) Power Rule for Derivatives
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \frac {\sqrt {x^2 - a^2} } {x^3} \rd x\) \(=\) \(\displaystyle \int \frac {\sqrt {z - a^2} \rd z} {2 z^{3/2} \sqrt z}\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \int \frac {\sqrt {z - a^2} \rd z} {z^2}\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac {-\sqrt {z - a^2} } z + \frac 1 2 \int \frac {\d z} {z \sqrt {z - a^2} } } + C\) Primitive of $\dfrac {\sqrt {a x + b} } {x^m}$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 4 \int \frac {2 x \rd x} {x^2 \sqrt {x^2 - a^2} } + C\) substituting for $z$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 2 \int \frac {\d x} {x \sqrt {x^2 - a^2} } + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 2 \paren {\frac 1 a \arcsec \size {\frac x a} } + C\) Primitive of $\dfrac 1 {x \sqrt {x^2 - a^2} }$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 {2 a} \arcsec \size {\frac x a} + C\) simplifying

$\blacksquare$


Also see


Sources