Primitive of Cosecant Function/Tangent Form/Proof 1
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Theorem
- $\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$
where $\tan \dfrac x 2 \ne 0$.
Proof
\(\ds \int \csc x \rd x\) | \(=\) | \(\ds -\ln \size {\csc x + \cot x} + C\) | Primitive of $\csc x$: Cosecant plus Cotangent Form | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {\frac 1 {\csc x + \cot x} } + C\) | Logarithm of Reciprocal | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {\frac 1 {\frac 1 {\sin x} + \frac {\cos x} {\sin x} } } + C\) | Definition of Cosecant and Definition of Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {\frac {\sin x} {1 + \cos x} } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {\tan \frac x 2} + C\) | Half Angle Formula for Tangent: Corollary $1$ |
$\blacksquare$