Primitive of Function of Arccosecant

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$\ds \int \map F {\arccsc \frac x a} \rd x = -a \int \map F u \size {\csc u} \cot u \rd u$

where $u = \arccsc \dfrac x a$.


First note that:

\(\ds u\) \(=\) \(\ds \arccsc \frac x a\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \csc u\) Definition of Arccosecant


\(\ds u\) \(=\) \(\ds \arccsc \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {-a} {\size x {\sqrt {x^2 - a^2} } }\) Derivative of Arccosecant Function: Corollary
\(\ds \leadsto \ \ \) \(\ds \int \map F {\arccsc \frac x a} \rd x\) \(=\) \(\ds \int \map F u \frac {\size x {\sqrt {x^2 - a^2} } } {-a} \rd u\) Primitive of Composite Function
\(\ds \) \(=\) \(\ds \int \map F u \frac {\size {a \csc u} {\sqrt {a^2 \csc^2 u - a^2} } } {-a} \rd u\) Definition of $x$
\(\ds \) \(=\) \(\ds \int \map F u \size {\csc u} \paren {-\sqrt {a^2 \csc^2 u - a^2} } \rd u\)
\(\ds \) \(=\) \(\ds \int \map F u \paren {-a} \size {\csc u} {\sqrt {\csc^2 u - 1} } \rd u\)
\(\ds \) \(=\) \(\ds \int \map F u \paren {-a} \size {\csc u} \cot u \rd u\) Difference of Squares of Cosecant and Cotangent
\(\ds \) \(=\) \(\ds -a \int \map F u \size {\csc u} \cot u \rd u\) Primitive of Constant Multiple of Function


Also see