Calculus/Integration techniques/Reduction Formula

A reduction formula is one that enables us to solve an integral problem by reducing it to a problem of solving an easier integral problem, and then reducing that to the problem of solving an easier problem, and so on.

For example, if we let

$I_{n}=\int x^{n}e^{x}dx$

Integration by parts allows us to simplify this to

$I_{n}=x^{n}e^{x}-n\int x^{n-1}e^{x}dx=$

$I_{n}=x^{n}e^{x}-nI_{n-1}$

which is our desired reduction formula. Note that we stop at

$I_{0}=e^{x}$ .

Similarly, if we let

$I_{n}=\int \sec ^{n}(\theta )d\theta$

then integration by parts lets us simplify this to

$I_{n}=\sec ^{n-2}(\theta )\tan(\theta )-(n-2)\int \sec ^{n-2}(\theta )\tan ^{2}(\theta )d\theta$

Using the trigonometric identity, $\tan ^{2}(\theta )=\sec ^{2}(\theta )-1$ , we can now write

$I_{n}$	$=\sec ^{n-2}(\theta )\tan(\theta )+(n-2)\left(\int \sec ^{n-2}(\theta )d\theta -\int \sec ^{n}(\theta )d\theta \right)$
	$=\sec ^{n-2}(\theta )\tan(\theta )+(n-2)\left(I_{n-2}-I_{n}\right)$

Rearranging, we get

$I_{n}={\frac {\sec ^{n-2}(\theta )\tan(\theta )}{n-1}}+{\frac {n-2}{n-1}}I_{n-2}$

Note that we stop at $n=1$ or 2 if $n$ is odd or even respectively.

As in these two examples, integrating by parts when the integrand contains a power often results in a reduction formula.