Reduction Formulae

Certain definite integrals with a repeated power (an index) can be evaluated by an index-reduction method instead of direct computation. This section obtains the values of

and also the value of the improper integral $\int_0^\infty e^{-x}x^n\,dx$ (used again in §9.7).

The method (3 steps). Step 1: identify the index (positive integer) $n$ in the integral. Step 2: name the integral $I_n$. Step 3: apply integration by parts to obtain an equation for $I_n$ in terms of $I_{n-1}$ or $I_{n-2}$ — this equation is the reduction formula.

Reduction Formula I. If $I_n=\displaystyle\int_0^{\pi/2}\sin^nx\,dx$, then $I_n=\dfrac{n-1}{n}I_{n-2}$, $n\ge2$.

Reduction Formula II. If $I_n=\displaystyle\int_0^{\pi/2}\cos^nx\,dx$, then $I_n=\dfrac{n-1}{n}I_{n-2}$, $n\ge2$.

Reduction Formula III. If $I_{m,n}=\displaystyle\int_0^{\pi/2}\sin^mx\cos^nx\,dx$, then $I_{m,n}=\dfrac{n-1}{m+n}I_{m,n-2}$, $n\ge2$.

Reduction Formula IV. If $I_{m,n}=\displaystyle\int_0^1x^m(1-x)^n\,dx$, then $I_{m,n}=\dfrac{n}{m+n+1}I_{m,n-1}$, $n\ge1$.

Closed forms from I and II (stated without proof, iterating down to the base case $I_0=\pi/2$ or $I_1=1$):

For instance $\displaystyle\int_0^{\pi/2}\cos^5x\,dx=\int_0^{\pi/2}\sin^5x\,dx=\dfrac{4}{5}\cdot\dfrac23\cdot1=\dfrac{8}{15}$, and $\displaystyle\int_0^{\pi/2}\sin^6x\,dx=\int_0^{\pi/2}\cos^6x\,dx=\dfrac56\cdot\dfrac34\cdot\dfrac12\cdot\dfrac\pi2=\dfrac{5\pi}{32}$.

Iterating Formula III gives, when $n$ is even and $m$ is even,

and, when $n$ is odd and $m$ is any positive integer,

$\int_0^{\pi/2}\sin^mx\cos^nx\,dx=\dfrac{(n-1)(n-3)\cdots2}{(m+n)(m+n-2)\cdots(m+2)}.$ …