Primitive of Function of Arccotangent

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$\ds \int \map F {\arccot \frac x a} \rd x = -a \int \map F u \csc^2 u \rd u$

where $u = \arccot \dfrac x a$.


First note that:

\(\ds u\) \(=\) \(\ds \arccot \frac x a\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \cot u\) Definition of Arccotangent


\(\ds u\) \(=\) \(\ds \arccot \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {-a} {a^2 + x^2}\) Derivative of Arccotangent Function: Corollary
\(\ds \leadsto \ \ \) \(\ds \int \map F {\arccot \frac x a} \rd x\) \(=\) \(\ds \int \map F u \frac {a^2 + x^2} {-a} \rd u\) Primitive of Composite Function
\(\ds \) \(=\) \(\ds \int \map F u \frac {a^2 + a^2 \cot^2 u} {-a} \rd u\) Definition of $x$
\(\ds \) \(=\) \(\ds \int \map F u a^2 \frac {1 + \cot^2 u} {-a} \rd u\)
\(\ds \) \(=\) \(\ds \int \map F u \paren {-a} \csc^2 u \rd u\) Difference of Squares of Cosecant and Cotangent
\(\ds \) \(=\) \(\ds -a \int \map F u \csc^2 u \rd u\) Primitive of Constant Multiple of Function


Also see