2.4.2—And use reduction formulae for the

A reduction formula expresses an integral I_n in terms of I_{n−1} or I_{n−2}, usually by integration by parts. Repeating the relation eventually reaches a base integral that can be evaluated directly.

Define the indexed integral and its limits clearly, derive the recurrence once, then apply it with the correct starting value. Track boundary terms and signs at each step.

For I_n=∫sin^n x dx on a fixed interval, integration by parts can relate I_n to I_{n−2}; repeated use reduces an even or odd power to I_0 or I_1.

A recurrence is not the final numerical answer: the base case and the index range are part of the proof.