Primitive of Square of Cotangent Function
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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: $16.20.$