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

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Theorem

$\ds \int \frac {\d x} {x \sqrt {x^n - a^n} } = \frac 2 {n \sqrt {a^n} } \arccos \sqrt {\frac {a^n} {x^n} }$


Proof

\(\ds u\) \(=\) \(\ds \sqrt {x^n - a^n}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {n x^{n - 1} } {2 \sqrt {x^n - a^n} }\) Derivative of Power, Chain Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {x \sqrt {x^n - a^n} }\) \(=\) \(\ds \int \frac {2 \sqrt {x^n - a^n} \rd u} {n x^{n - 1} x \sqrt {x^n - a^n} }\) Integration by Substitution
\(\ds \) \(=\) \(\ds \int \frac {2 \rd u} {n \paren {u^2 + a^n} }\) completing substitution and simplifying
\(\ds \) \(=\) \(\ds \frac 2 n \int \frac {\d u} {\paren {u^2 + \paren {\sqrt {a^n} }^2} }\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 2 n \paren {\frac 1 {\sqrt {a^n} } \arctan \frac u {\sqrt {a^n} } } + C\) Primitive of $\dfrac 1 {x^2 + a^2}$
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \frac 2 {n \sqrt {a^n} } \arctan \frac {\sqrt {x^n - a^n} } {\sqrt {a^n} } + C\) substituting for $u$


Now:

\(\ds y\) \(=\) \(\ds \arctan \frac {\sqrt {x^n - a^n} } {\sqrt {a^n} }\)
\(\ds \leadsto \ \ \) \(\ds \tan y\) \(=\) \(\ds \frac {\sqrt {x^n - a^n} } {\sqrt {a^n} }\) Definition of Real Arctangent
\(\ds \leadsto \ \ \) \(\ds \tan^2 y\) \(=\) \(\ds \frac {x^n - a^n} {a^n}\) squaring both sides
\(\ds \leadsto \ \ \) \(\ds \tan^2 y\) \(=\) \(\ds \frac {x^n} {a^n} - 1\)
\(\ds \leadsto \ \ \) \(\ds 1 + \tan^2 y\) \(=\) \(\ds \frac {x^n} {a^n}\)
\(\ds \leadsto \ \ \) \(\ds \sec^2 y\) \(=\) \(\ds \frac {x^n} {a^n}\) Difference of Squares of Secant and Tangent
\(\ds \leadsto \ \ \) \(\ds \cos^2 y\) \(=\) \(\ds \frac {a^n} {x^n}\) Definition of Secant Function
\(\ds \leadsto \ \ \) \(\ds \cos y\) \(=\) \(\ds \frac {\sqrt {a^n} } {\sqrt {x^n} }\)
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds \arccos \frac {\sqrt {a^n} } {\sqrt {x^n} }\) Definition of Real Arccosine
\(\ds \leadsto \ \ \) \(\ds \int \frac {\d x} {x \sqrt {x^n - a^n} }\) \(=\) \(\ds \frac 2 {n \sqrt {a^n} } \arccos \sqrt {\frac {a^n} {x^n} }\) substituting in $(1)$

$\blacksquare$


Sources

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