Primitive of Cosecant Function/Cosecant plus Cotangent Form/Proof
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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 |
$\blacksquare$