Reduction Formula for Definite Integral of Power of Cosine

Theorem
Let $n \in \Z_{> 0}$ be a positive integer.

Let $I_n$ be defined as:
 * $\ds I_n = \int_0^{\frac \pi 2} \cos^n x \rd x$

Then $\sequence {I_n}$ is a decreasing sequence of real numbers which satisfies:
 * $n I_n = \paren {n - 1} I_{n - 2}$

Thus:
 * $I_n = \dfrac {n - 1} n I_{n - 2}$

is a reduction formula for $I_n$.

Proof
From Shape of Cosine Function:
 * $\forall x \in \closedint 0 {\dfrac \pi 2}: 0 \le \cos x \le 1$

So, on the same interval:
 * $0 \le \cos^{n + 1} x \le \cos^n x$

Therefore:
 * $\forall n \in \N: 0 < I_{n + 1} < I_n$

From Reduction Formula for Integral of Power of Cosine:


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

Thus:

Also see

 * Reduction Formula for Definite Integral of Power of Sine