Reduction formulas

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Integration by parts can relate a power-n integral to a lower power. Repeat the recurrence until reaching a known base case. Formulas depend on the family and parameters.

Step-by-step example

1. Separate one sine and integrate by parts.

   $$u=\sin^{n-1}x,\quad dv=\sin x\,dx$$

2. Use cos²x = 1−sin²x and isolate Iₙ.

   $$I_n=-\frac{\sin^{n-1}x\cos x}{n}+\frac{n-1}{n}I_{n-2}$$

3. For n = 3 use I₁ = −cos x.

   $$I_3=-\frac13\sin^2x\cos x-\frac23\cos x+C$$