Integration by parts is a direct consequence of the product rule for the derivative and teh fundamental theorem of calculus. Recall that the product rule says that \[(fg)'=f'g+fg'\]Solving for we have \[fg'=(fg)'-f'g\] By integrating both sides we have \[\int f(x)g'(x)\,dx=f(x)g(x)-\int g(d) f'(x)\,dx\] Denoting and , the above integral is more popular in the form \[\int u\,dv=uv-\int v\,du\]