Primitive of Cube of Cotangent of a x

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Theorem

$\displaystyle \int \cot^3 a x \rd x = \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C$


Proof

\(\displaystyle \int \cot^3 x \rd x\) \(=\) \(\displaystyle \int \cot a x \cot^2 a x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \int \cot a x \paren {\csc^2 a x - 1} \rd x\) Difference of Squares of Cosecant and Cotangent
\(\displaystyle \) \(=\) \(\displaystyle \int \cot a x \csc^2 a x \rd x - \int \cot a x \rd x\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\cot^2 a x} {2 a} - \int \cot a x \rd x + C\) Primitive of $\cot^n a x \csc^2 a x$: $n = 1$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\cot^2 a x} {2 a} - \paren {\frac {\ln \size {\sin a x} } a} + C\) Primitive of $\cot a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C\) simplifying

$\blacksquare$


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