Primitive of Square of Cotangent Function

Theorem

$\ds \int \cot^2 x \rd x = -\cot x - x + C$

where $C$ is an arbitrary constant.


Proof

\(\ds \int \cot^2 x \rd x\) \(=\) \(\ds \int \paren {\csc^2 x - 1} \rd x\) Difference of Squares of Cosecant and Cotangent
\(\ds \) \(=\) \(\ds \int \csc^2 x \rd x + \int \paren {-1} \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds -\cot x + C + \int \paren {-1} \rd x\) Primitive of Square of Cosecant Function
\(\ds \) \(=\) \(\ds -\cot x - x + C\) Primitive of Constant

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.20$
  • 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.20.$
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