Primitive of Power of Cosecant of a x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \csc^n a x \rd x = \frac{-\csc^{n - 2} a x \cot a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \csc^{n - 2} a x \rd x$

where $n \ne -1$.


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds \csc^{n - 2} a x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds -a \paren {n - 2} \csc^{n - 3} a x \csc a x \cot a x\) Derivative of Power, Derivative of $\csc$, Chain Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds -a \paren {n - 2} \csc^{n - 2} a x \cot a x\)


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \csc^2 a x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {-\cot a x} a\) Primitive of $\csc^2 a x$


Then:

\(\ds \int \csc^n a x \rd x\) \(=\) \(\ds \int \csc^{n - 2} a x \csc^2 a x \rd x\)
\(\ds \) \(=\) \(\ds \csc^{n - 2} a x \paren {\frac {-\cot a x} a} - \int \paren {\frac {-\cot a x} a} \paren {-a \paren {n - 2} \csc^{n - 2} a x \cot a x } \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {-\csc^{n - 2} a x \cot a x} a - \paren {n - 2} \int \cot^2 a x \csc^{n - 2} a x \rd x\) simplifying
\(\ds \) \(=\) \(\ds \frac {-\csc^{n - 2} a x \cot a x} a - \paren {n - 2} \int \paren {\csc^2 a x - 1} \csc^{n - 2} a x \rd x\) Difference of $\csc^2$ and $\cot^2$
\(\ds \) \(=\) \(\ds \frac {-\csc^{n - 2} a x \cot a x} a - \paren {n - 2} \int \csc^n a x \rd x\) Linear Combination of Primitives
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {n - 2} \int \csc^{n - 2} a x \rd x\)
\(\ds \paren {n - 1} \int \csc^n a x \rd x\) \(=\) \(\ds \frac {-\csc^{n - 2} a x \cot a x} a + \paren {n - 2} \int \csc^{n - 2} a x \rd x\) gathering terms
\(\ds \int \csc^n a x \rd x\) \(=\) \(\ds \frac{-\csc^{n - 2} a x \cot a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \csc^{n - 2} a x \rd x\) dividing by $n - 1$

$\blacksquare$


Also see


Sources