Primitive of Cosecant Function/Tangent Form/Proof 1
Jump to navigation
Jump to search
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 $1$ |
$\blacksquare$