Question

How to find what is the first function in by parts integration?

Answer 1

Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula. If u = u(x) and du = u′(x) dx, while v = v(x) and dv = v′(x) dx, then integration by parts states that:

{\displaystyle {\begin{aligned}\int _{a}^{b}u(x)v'(x)\,dx&=[u(x)v(x)]_{a}^{b}-\int _{a}^{b}u'(x)v(x)dx\\&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx\end{aligned}}}

or more compactly:

{\displaystyle \int u\,dv=uv-\int v\,du.\!}