Primitive of Cotangent of a x/Examples/Half x

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 $\text {XIV}$: $3$.