Primitive of Cosecant Function/Cosecant minus Cotangent Form

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Theorem

$\ds \int \csc x \rd x = \ln \size {\csc x - \cot x} + C$

where $\csc x - \cot x \ne 0$.


Proof

\(\ds \int \csc x \rd x\) \(=\) \(\ds \ln \size {\tan \frac x 2} + C\) Primitive of $\csc x$: Tangent Form
\(\ds \) \(=\) \(\ds \ln \size {\frac {1 - \cos x} {\sin x} } + C\) Half Angle Formula for Tangent: Corollary $2$
\(\ds \) \(=\) \(\ds \ln \size {\frac 1 {\sin x} - \frac {\cos x} {\sin x} } + C\)
\(\ds \) \(=\) \(\ds \ln \size {\csc x - \cot x} + C\) Definition of Cosecant and Definition of Cotangent

$\blacksquare$


Sources

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