Primitive of Cube of Cosecant of a x

From ProofWiki
Jump to navigation Jump to search

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


Sources