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

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

$I_n=x^n{e}^x-n{I}_{n-1}$

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

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

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.