Primitive of Cosecant Function/Tangent Form/Proof 2
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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
| \(\ds \int \csc x \rd x\) | \(=\) | \(\ds \int \frac 1 {\sin x} \rd x\) | Cosecant is Reciprocal of Sine |
We make the Weierstrass Substitution:
| \(\ds u\) | \(=\) | \(\ds \tan \frac x 2\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin x\) | \(=\) | \(\ds \frac {2 u} {u^2 + 1}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d u}\) | \(=\) | \(\ds \frac 2 {u^2 + 1}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \frac 1 {\sin x} \rd x\) | \(=\) | \(\ds \int \frac {u^2 + 1} {2 u} \frac 2 {u^2 + 1} \rd u\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac 1 u \rd u\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size u + C\) | Primitive of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds \ln \size {\tan \frac x 2} + C\) | substituting back for $u$ |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Useful substitutions: Example $1$.