recursive integration

Recursive integration is the broad habit of solving an integral by relating it to itself or to a simpler version of itself. Reduction formulas are one species of it, but the family is larger: sometimes integration by parts brings the original integral back on the right-hand side, and you solve for it algebraically as if it were an unknown. The label captures the spirit of leaning on the structure of the problem rather than finding a fresh antiderivative.

The classic case is the integral of e^x cos(x) dx. Integrate by parts once and you get a new integral of e^x sin(x); integrate that by parts and the e^x cos(x) integral reappears with a sign. Calling the original I, you arrive at an equation like I = (something) - I, which you simply solve: 2I = (something), so I equals half of it. No base case, no marching down — the recursion closes on itself in one loop. Other recursive setups do march, like reduction formulas, terminating at a known small case.

This way of thinking generalizes far beyond integrals: it is the same logic as a recurrence relation for series coefficients in the power-series and Frobenius methods, where each coefficient is defined from earlier ones. The honest caveat is that the algebraic-loop trick only works when the integral genuinely returns to itself (up to a constant factor); if the parts choice keeps producing brand-new integrals, you are not recursing, you are wandering, and you should pick a different split.