Reduction Formulae for Definite Integrals of sin^n x and cos^n x

For the special family of integrals $\int_0^{\pi/2}\sin^n x\,dx$ and $\int_0^{\pi/2}\cos^n x\,dx$ (products of the same power of sine or cosine over a quarter period), there is a standard shortcut called a reduction formula that gives the value directly from $n$, without repeated integration by parts:

The pattern is: start with $\frac{n-1}{n}$ and keep multiplying by the next fraction obtained by subtracting $2$ from both the numerator's two terms and the denominator's two terms, all the way down to $\frac23$ if $n$ is odd (no extra factor), or down to $\frac12$ followed by one extra factor of $\pi/2$ if $n$ is even.

Reducing cosine to sine. By Property V (Section 4.2.1) with the reflection $x\to \pi/2-x$,

so the same reduction formula, with the same value of $n$, gives $\int_0^{\pi/2}\cos^n x\,dx$ as well — there is no need to memorise a second formula.

Worked check for $n=7$ (odd). $\displaystyle\int_0^{\pi/2}\sin^7x\,dx=\frac{6}{7}\cdot\frac{4}{5}\cdot\frac{2}{3}=\frac{6\cdot4\cdot2}{7\cdot5\cdot3}=\frac{16}{35}$ (the chain stops at $\frac23$ since $7$ is odd, with no leftover $\pi/2$ factor). …