Primitive of Cotangent of a x/Examples/Half x
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Example of Use of Primitive of $\cot a x$
- $\ds \int \cot \dfrac x 2 \rd x = 2 \ln \size {\sin \dfrac 1 2} + C$
Proof
From Primitive of $\cot a x$:
- $\ds \int \cot a x \rd x = \frac {\ln \size {\sin a x} } a + C$
Then:
\(\ds \ds \int \cot \dfrac x 2 \rd x\) | \(=\) | \(\ds \dfrac 1 {1 / 2} \ln \size {\sin \dfrac 1 2} + 2\) | setting $a \gets \dfrac 1 2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \ln \size {\sin \dfrac 1 2} + C\) |
The result follows by setting $a = \dfrac 1 2$.
$\blacksquare$
Proof
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $3$.