Primitive of Square of Cosecant of a x
Theorem
- $\ds \int \csc^2 a x \rd x = -\frac {\cot a x} a + C$
Proof
| \(\ds \int \csc^2 x \rd x\) | \(=\) | \(\ds -\cot x + C\) | Primitive of $\csc^2 x$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \csc^2 a x \rd x\) | \(=\) | \(\ds \frac 1 a \paren {-\cot a x} + C\) | Primitive of Function of Constant Multiple | ||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac {\cot a x} a + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\sin^2 a x$
- Primitive of $\cos^2 a x$
- Primitive of $\tan^2 a x$
- Primitive of $\cot^2 a x$
- Primitive of $\sec^2 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csc a x$: $14.462$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $91$.
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 17$: Tables of Special Indefinite Integrals: $(23)$ Integrals Involving $\csc a x$: $17.23.2.$