Primitive of Cosecant Function/Cosecant minus 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
\(\ds \int \csc x \rd x\) | \(=\) | \(\ds \ln \size {\tan \frac x 2} + C\) | 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
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Useful substitutions: Example $1$.