Integration By Parts - Recursive Integration By Parts

Recursive Integration By Parts

Integration by parts can often be applied recursively on the term to provide the following formula

Here, is the first derivative of and is the second derivative. Further, is a notation to describe its nth derivative with respect to the independent variable. Another notation approved in the calculus theory has been adopted:

There are n + 1 integrals.

Note that the integrand above (uv) differs from the previous equation. The dv factor has been written as v purely for convenience.

The above mentioned form is convenient because it can be evaluated by differentiating the first term and integrating the second (with a sign reversal each time), starting out with uv₁. It is very useful especially in cases when u(k+1) becomes zero for some k + 1. Hence, the integral evaluation can stop once the u(k) term has been reached.