Reduction Formula for Integral of Power of Cosine

Theorem

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

Then:

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

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

Corollary

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

Proof

With a view to expressing the problem in the form:

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

let:

\(\ds u\)

\(=\)

\(\ds \cos^{n - 1} x\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\rd u} {\rd x}\)

\(=\)

\(\ds -\paren {n - 1} \cos ^{n - 2} x \sin x\)

Chain Rule for Derivatives, Derivative of Cosine Function, Derivative of Power

and let:

\(\ds \frac {\rd v} {\rd x}\)

\(=\)

\(\ds \cos x\)

\(\ds \leadsto \ \ \)

\(\ds v\)

\(=\)

\(\ds \sin x\)

Primitive of Cosine Function

Then:

\(\ds \int \cos^n x \rd x\)

\(=\)

\(\ds \int \cos^{n - 1} x \cos x \rd x\)

\(\ds \)

\(=\)

\(\ds \cos^{n - 1} x \sin x - \int \sin x \paren {-\paren {n - 1} \cos^{n - 2} x \sin x} \rd x\)

Integration by Parts

\(\ds \)

\(=\)

\(\ds \cos^{n - 1} x \sin x + \int \paren {n - 1} \cos^{n - 2} x \sin^2 x \rd x\)

rearranging

\(\ds \)

\(=\)

\(\ds \cos^{n - 1} x \sin x + \int \paren {n - 1} \cos^{n - 2} x \paren {1 - \cos^2 x} \rd x\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds \cos^{n - 1} x \sin x + \paren {n - 1} \int \cos^{n - 2} x \rd x - \int \paren {n - 1} \cos^n x \rd x\)

Linear Combination of Primitives

\(\ds \leadsto \ \ \)

\(\ds n \int \cos^n x \rd x\)

\(=\)

\(\ds \cos^{n - 1} x \sin x + \paren {n - 1} \int \cos^{n - 2} x \rd x\)

rearranging

\(\ds \leadsto \ \ \)

\(\ds \int \cos^n x \rd x\)

\(=\)

\(\ds \frac {\cos^{n - 1} x \sin x} n + \frac {n - 1} n \int \cos^{n - 2} x \rd x\)

dividing both sides by $n$

$\blacksquare$

Also see

• Reduction Formula for Integral of Power of Sine
• Reduction Formula for Integral of Power of Tangent