Primitive of Cube of Cosecant of a x
Theorem
- $\ds \int \csc^3 a x \rd x = \frac {-\csc a x \cot a x} {2 a} + \frac 1 {2 a} \ln \size {\tan \dfrac {a x} 2} + C$
Proof
| \(\ds \int \csc^3 x \rd x\) | \(=\) | \(\ds \frac {-\csc a x \cot a x} {2 a} + \frac 1 2 \int \csc a x \rd x\) | Primitive of $\csc^n a x$ where $n = 3$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-\csc a x \cot a x} {2 a} + \frac 1 2 \paren {\frac 1 a \ln \size {\tan \dfrac {a x} 2} }\) | Primitive of $\csc a x$: Tangent Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {-\csc a x \cot a x} {2 a} + \frac 1 {2 a} \ln \size {\tan \dfrac {a x} 2} + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\sin^3 a x$
- Primitive of $\cos^3 a x$
- Primitive of $\tan^3 a x$
- Primitive of $\cot^3 a x$
- Primitive of $\sec^3 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csc a x$: $14.463$