[DEFINITIONS] Reduction formulae

Reduction formula

A reduction formula expresses an integral $I_n$ in terms of $I_{n-k}$ for a smaller index, typically by integration by parts.

Here and below, power indices are non-negative integers. The step $k$ is $1$ for $x^n e^x$ . For trigonometric powers it is $2$ , because $\sin^2 x+\cos^2 x=1$ or $\sec^2 x=1+\tan^2 x$ returns a copy of $I_n$ that is then collected.

[1]:Powers of $x$ times $e^x$
[2]:Sine and cosine powers on $\bigl[0,\frac{\pi}{2}\bigr]$
[3]:Secant powers on $\bigl[0,\frac{\pi}{4}\bigr]$
[4]:Tangent powers on $\bigl[0,\frac{\pi}{4}\bigr]$
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Definitions from Reduction formulae

Reduction formula

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