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
![{\displaystyle I_{n}=\int x^{n}e^{x}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/046f726f22515986fefcf77225ca3f05662eab52)
Integration by parts allows us to simplify this to
![{\displaystyle I_{n}=x^{n}e^{x}-n\int x^{n-1}e^{x}dx=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a095dcc83877071f0423eb3657523e263b8773d)
![{\displaystyle I_{n}=x^{n}e^{x}-nI_{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20b9a86d75a0c2e49ea08d1d90a774896907bb24)
which is our desired reduction formula. Note that we stop at
.
Similarly, if we let
![{\displaystyle I_{n}=\int \sec ^{n}(\theta )d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/402061c3894dae39a509f4b3e3ca1bd63f50bff7)
then integration by parts lets us simplify this to
![{\displaystyle I_{n}=\sec ^{n-2}(\theta )\tan(\theta )-(n-2)\int \sec ^{n-2}(\theta )\tan ^{2}(\theta )d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6518eeaf1157b7ee26a14cecabc3127a8eae9e)
Using the trigonometric identity,
, we can now write
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Rearranging, we get
![{\displaystyle I_{n}={\frac {\sec ^{n-2}(\theta )\tan(\theta )}{n-1}}+{\frac {n-2}{n-1}}I_{n-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f9d9acd2fde9b02d9d27851bff9709ee68bfec1)
Note that we stop at
or 2 if
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.