The next most common integration technique is based on the idea of the product rule. Suppose that u and v are both functions of x. Then
Now, if we rearrange this, we get u dv = d(uv) - v du. Integrating both sides produces the formula for integration by parts:
In practice, the trick is always how to identify u and dv in the integral given so that the integrals on the right hand side are simpler. As a general rule, we want to choose u to be a function that, when we take a derivative to get du, is simpler.
