Primitive of Reciprocal of Power of Root of x squared minus a squared

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Theorem

$\ds \int \dfrac {\d x} {\paren {\sqrt {x^2 - a^2} }^n} = \dfrac {x \paren {\sqrt {x^2 - a^2} }^{2 - n} } {\paren {2 - n} a^2} - \dfrac {n - 3} {\paren {n - 2} a^2} \int \dfrac {\d x} {\paren {\sqrt {x^2 - a^2} }^{n - 2} }$

for $n \ne 2$.


Proof

Let:

\(\ds x\) \(=\) \(\ds a \cosh \theta\)
\(\text {(1)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d \theta}\) \(=\) \(\ds a \sinh \theta\) Derivative of Hyperbolic Cosine


Also:

\(\ds x\) \(=\) \(\ds a \cosh \theta\)
\(\ds \leadsto \ \ \) \(\ds x^2 - a^2\) \(=\) \(\ds a^2 \cosh^2 \theta - a^2\)
\(\ds \) \(=\) \(\ds a^2 \paren {\cosh^2 \theta - 1}\)
\(\ds \) \(=\) \(\ds a^2 \sinh^2 \theta\) Difference of Squares of Hyperbolic Cosine and Sine
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \paren {\sqrt {x^2 - a^2} }^n\) \(=\) \(\ds a^n \sinh^n \theta\)


and:

\(\ds x\) \(=\) \(\ds a \cosh \theta\)
\(\text {(3)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \theta\) \(=\) \(\ds \cosh^{-1} \frac x a\) Definition 1 of Real Inverse Hyperbolic Cosine


Thus:

\(\ds \int \dfrac {\d x} {\paren {\sqrt {x^2 - a^2} }^n}\) \(=\) \(\ds \int \dfrac {a \sinh \theta \rd \theta} {\paren {\sqrt {x^2 - a^2} }^n}\) Integration by Substitution from $(1)$
\(\ds \) \(=\) \(\ds \int \dfrac {a \sinh \theta \rd \theta} {a^n \sinh^n \theta}\) substituting for $\paren {\sqrt {x^2 - a^2} }^n$ from $(2)$
\(\ds \) \(=\) \(\ds \dfrac 1 {a^{n - 1} } \int \dfrac {\d \theta} {\sinh^{n - 1} \theta}\) Primitive of Constant Multiple of Function and simplification
\(\ds \) \(=\) \(\ds \dfrac 1 {a^{n - 1} } \paren {\frac {-\cosh \theta} {\paren {n - 2} \sinh^{n - 2} \theta} - \frac {n - 3} {n - 2} \int \frac {\d \theta} {\sinh^{n - 3} \theta} }\) Primitive of $\dfrac 1 {\sinh^{n - 1} \theta}$ for $n \ne 2$
\(\ds \) \(=\) \(\ds \frac {-a \cosh \theta} {\paren {n - 2} a^{n - 2} \sinh^{n - 2} \theta} - \frac {n - 3} {\paren {n - 2} a^2} \int \frac {a \sinh \theta \rd \theta} {a^{n - 2} \sinh^{n - 2} \theta}\) rearranging
\(\ds \) \(=\) \(\ds \frac {-x} {\paren {n - 2} \paren {\sqrt {x^2 - a^2} }^{n - 2} } - \frac {n - 3} {\paren {n - 2} a^2} \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^{n - 2} }\) substituting for $\sinh \theta$ and $\cosh \theta$
\(\ds \) \(=\) \(\ds \dfrac {x \paren {\sqrt {x^2 - a^2} }^{2 - n} } {\paren {2 - n} a^2} - \frac {n - 3} {\paren {n - 2} a^2} \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^{n - 2} }\) rearranging into given form

$\blacksquare$


Also see

For $n = -2$, use Primitive of $\dfrac 1 {x^2 - a^2}$.


Sources

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