A reduction formula

is a mathematical technique, mainly used in integration, to reduce a complex integral to a simpler one, typically involving a lower power or simpler function. It’s especially useful in definite and indefinite integrals of powers, trigonometric functions, and exponential or logarithmic forms.

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🔹 General Format of a Reduction Formula:

A reduction formula expresses:

In=∫fn(x) dxI_n = \int f_n(x) \, dxIn​=∫fn​(x)dx

in terms of:

In−1=∫fn−1(x) dxI_{n-1} = \int f_{n-1}(x) \, dxIn−1​=∫fn−1​(x)dx

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🔸 Common Examples of Reduction Formulas

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1. Power of sine:

In=∫sin⁡nx dxI_n = \int \sin^n x \, dxIn​=∫sinnxdx

Reduction formula:

∫sin⁡nx dx=−1nsin⁡n−1xcos⁡x+n−1n∫sin⁡n−2x dx\int \sin^n x \, dx = -\frac{1}{n} \sin^{n-1}x \cos x + \frac{n-1}{n} \int \sin^{n-2}x \, dx∫sinnxdx=−n1​sinn−1xcosx+nn−1​∫sinn−2xdx

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2. Power of cosine:

∫cos⁡nx dx=1ncos⁡n−1xsin⁡x+n−1n∫cos⁡n−2x dx\int \cos^n x \, dx = \frac{1}{n} \cos^{n-1}x \sin x + \frac{n-1}{n} \int \cos^{n-2}x \, dx∫cosnxdx=n1​cosn−1xsinx+nn−1​∫cosn−2xdx

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3. Exponential and polynomial:

∫xnex dx\int x^n e^x \, dx∫xnexdx

Reduction formula:

∫xnex dx=xnex−n∫xn−1ex dx\int x^n e^x \, dx = x^n e^x – n \int x^{n-1} e^x \, dx∫xnexdx=xnex−n∫xn−1exdx

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4. Power of log:

∫(ln⁡x)ndx\int (\ln x)^n dx∫(lnx)ndx

Reduction formula:

∫(ln⁡x)ndx=x(ln⁡x)n−n∫(ln⁡x)n−1dx\int (\ln x)^n dx = x (\ln x)^n – n \int (\ln x)^{n-1} dx∫(lnx)ndx=x(lnx)n−n∫(lnx)n−1dx

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Would you like a step-by-step derivation or examples using any of these?

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