Reduction formulas are shortcuts that turn a difficult integral into a simpler integral of the same family. They are especially useful when an integrand contains a power such as sin^n x, cos^n x, x^n e^x, or powers of logarithms. Instead of starting from scratch each time, a reduction formula lets you step down from a large exponent to a smaller one.
This makes long integration problems more organized and less error prone.
Most reduction formulas come from integration by parts, which rewrites an integral using a product of two functions. The key idea is to choose u and dv so that the new integral has a lower power or a simpler form than the original. For example, I_n = ∫sin^n x dx can be related to I_{n - 2}, so each step reduces the exponent by 2.
After repeated steps, the process ends at a basic integral such as I_0 = ∫1 dx = x or I_1 = ∫sin x dx = -cos x.
Key Facts
- Integration by parts: ∫u dv = uv - ∫v du.
- A reduction formula expresses I_n in terms of I_{n - 1}, I_{n - 2}, or another simpler related integral.
- For I_n = ∫sin^n x dx, one reduction formula is I_n = -sin^{n - 1} x cos x / n + ((n - 1) / n) I_{n - 2}.
- For J_n = ∫cos^n x dx, one reduction formula is J_n = sin x cos^{n - 1} x / n + ((n - 1) / n) J_{n - 2}.
- For K_n = ∫x^n e^x dx, one reduction formula is K_n = x^n e^x - nK_{n - 1}.
- A complete answer must continue the recurrence until it reaches a base case and then include the constant of integration C.
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