Primitive of Cosecant Function/Cosecant plus 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

Let:

\(\ds u\) \(=\) \(\ds \cot x + \csc x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac \d {\d x} \cot x + \frac \d {\d x} \csc x\) Linear Combination of Derivatives
\(\ds \) \(=\) \(\ds -\csc^2 x + \frac \d {\d x} \csc x\) Derivative of Cotangent Function
\(\ds \) \(=\) \(\ds -\csc^2 x - \csc x \cot x\) Derivative of Cosecant Function
\(\ds \) \(=\) \(\ds -\csc x \paren {\csc x + \cot x}\) factorising


Then:

\(\ds \int \csc x \rd x\) \(=\) \(\ds \int \frac {\csc x \paren {\csc x + \cot x} } {\csc x + \cot x} \rd x\) multiplying top and bottom by $\csc x + \cot x$
\(\ds \) \(=\) \(\ds -\int \frac {-\csc x \paren {\csc x + \cot x} } {\csc x + \cot x} \rd x\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds -\ln \size {\csc x + \cot x} + C\) Primitive of Function under its Derivative

$\blacksquare$


Sources