0.11: Integration by Parts
Another integration technique makes use of the product rule for differentiation. Since
\[(fg)'=f'g+fg',\nonumber \]
we have
\[f'g=(fg)'-fg'.\nonumber \]
Therefore,
\[\int f'(x)g(x)dx=f(x)g(x)-\int f(x)g'(x)dx.\nonumber \]
Commonly, the above integral is done by writing
\[\begin{array}{cc}u=g(x)&dv=f'(x)dx \\ du=g'(x)dx&v=f(x).\end{array}\nonumber \]
Then, the formula to be memorized is
\[\int udv=uv-\int vdu.\nonumber \]