Primitive of Function of Arccosecant

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Theorem

$\displaystyle \int F \left({\operatorname{arccsc} \frac x a}\right) \ \mathrm d x = -a \int F \left({u}\right) \left\vert{\csc u}\right\vert \cot u \ \mathrm d u$

where $u = \operatorname{arccsc} \dfrac x a$.


Proof

First note that:

\(\displaystyle u\) \(=\) \(\displaystyle \operatorname{arccsc} \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle a \csc u\) Definition of Arccosecant


Then:

\(\displaystyle u\) \(=\) \(\displaystyle \operatorname{arccsc} \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac{\mathrm d u} {\mathrm d x}\) \(=\) \(\displaystyle \frac {-a} {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } }\) Derivative of Arccosecant Function: Corollary
\(\displaystyle \implies \ \ \) \(\displaystyle \int F \left({\operatorname{arccsc} \frac x a}\right) \ \mathrm d x\) \(=\) \(\displaystyle \int F \left({u}\right) \ \frac {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } } {-a} \ \mathrm d u\) Primitive of Composite Function
\(\displaystyle \) \(=\) \(\displaystyle \int F \left({u}\right) \ \frac {\left\vert{a \csc u}\right\vert {\sqrt {a^2 \csc^2 u - a^2} } } {-a} \ \mathrm d u\) Definition of $x$
\(\displaystyle \) \(=\) \(\displaystyle \int F \left({u}\right) \ \left\vert{\csc u}\right\vert \left({-\sqrt {a^2 \csc^2 u - a^2} }\right) \ \mathrm d u\)
\(\displaystyle \) \(=\) \(\displaystyle \int F \left({u}\right) \ \left({-a}\right) \left\vert{\csc u}\right\vert {\sqrt {\csc^2 u - 1} } \ \mathrm d u\)
\(\displaystyle \) \(=\) \(\displaystyle \int F \left({u}\right) \ \left({-a}\right) \left\vert{\csc u}\right\vert \cot u \ \mathrm d u\) Difference of Squares of Cosecant and Cotangent
\(\displaystyle \) \(=\) \(\displaystyle -a \int F \left({u}\right) \ \left\vert{\csc u}\right\vert \cot u \ \mathrm d u\) Primitive of Constant Multiple of Function

$\blacksquare$



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