Reduction Formula for Integral of Power of Cosine/Corollary

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Corollary to Reduction Formula for Integral of Power of Cosine

Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

Let $a \in \R_{\ne 0}$ be a non-zero real number

Then:

$\ds \int \cos^n a x \rd x = \dfrac {\cos^{n - 1} a x \sin a x} {a n} + \dfrac {n - 1} n \int \cos^{n - 2} a x \rd x$

is a reduction formula for $\ds \int \cos^n a x \rd x$.


Proof

\(\ds \int \cos^n a x \rd x\) \(=\) \(\ds \frac 1 a \int \cos^n a x \map \rd {a x}\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \frac 1 a \paren {\frac {\cos^{n - 1} a x \sin a x} n + \frac {n - 1} n \int \cos^{n - 2} a x \map \rd {a x} }\) Reduction Formula for Integral of Power of Cosine
\(\ds \) \(=\) \(\ds \frac {\cos^{n-1} a x \sin a x} {a n} + \frac 1 a \paren {a \frac {n - 1} n \int \cos^{n - 2} a x \rd x}\) Primitive of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \frac {\cos^{n-1} a x \sin a x} {a n} + \frac {n - 1} n \int \cos^{n - 2} a x \rd x\) simplification

$\blacksquare$


Also see


Sources