Here’s the simple concept behind integration by parts as described in an earlier post. Integration by Parts simply what I call the reverse-product rule. Here’s why.
Let and be functions of . Using product rule, when we differentiate , we differentiate the first term *times* keep the second constant *plus* differentiate the second term *times* keep the first constant. That is,
Notationally, since and , we rewrite the above identity as
Let’s integrate with respect to on both sides. On the LHS, we are integrating what we get after differentiating. Since they cancel each other out, we get on the LHS. On the RHS, we get the integrals of each chunk added together, that is,
Subtracting by on both sides and switching the LHS with the RHS, we get
which is the famous integration by parts formula.
Hope this was insightful on how this technique is nothing more than reversing the product rule!