Primitive of Cotangent Function/Proof

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
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