Primitive of Cotangent Function/Proof
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Theorem
- $\ds \int \cot x \rd x = \ln \size {\sin x} + C$
where $\sin x \ne 0$.
Proof
\(\ds \int \cot x \rd x\) | \(=\) | \(\ds \int \frac {\cos x} {\sin x} \rd x\) | Definition of Real Cotangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {\sin x}'} {\sin x} \rd x\) | Derivative of Sine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln \size {\sin x} + C\) | Primitive of Function under its Derivative |
$\blacksquare$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration