what is reduction formula in integration?explain it

Dear Krish Mishra,

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

Integration by parts allows us to simplify this to

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

.

Similarly, if we let

then integration by parts lets us simplify this to

Using the trigonometric identity, tan²θ = sec²θ − 1, we can now write

Rearranging, we get

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.

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