# Calculus/Integration techniques/Reduction Formula

 ← Integration techniques/Tangent Half Angle Calculus Integration techniques/Irrational Functions → 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

$I_n = x^ne^x - n\int x^{n-1}e^x\,dx=$
$I_n = x^ne^x - nI_{n-1} \,\!$

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

$\begin{matrix} I_n & = & \sec^{n-2}\theta \tan \theta & + (n-2) \left( \int \sec^{n-2} \theta \, d\theta - \int \sec^n \theta \, d\theta \right) \\ & = & \sec^{n-2}\theta \tan \theta & + (n-2) \left( I_{n-2} - I_n \right) \\ \end{matrix}$

Rearranging, we get

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

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.

 ← Integration techniques/Tangent Half Angle Calculus Integration techniques/Irrational Functions → Integration techniques/Reduction Formula