Integration by Parts
Including Repeated Application
Trading one integral for a simpler one — the product rule read backwards
1. Derivation — The Product Rule Read Backwards
Starting Point: The Product Rule
Recall the product rule for derivatives. If $f(x)$ and $g(x)$ are differentiable functions, then:
Step 1: Integrate Both Sides
Integrate every term with respect to $x$. The left side "undoes" the derivative:
Step 2: Rearrange
Solve for the second integral:
Two Notational Forms
Let $u = f(x)$, $dv = g'(x)\,dx$
Then $du = f'(x)\,dx$, $v = g(x)$
This is the form you will use most often.
Useful when you want to see the functions and their derivatives explicitly.
What Does the Formula Actually Do?
Integration by parts trades one integral for another, plus a boundary product $uv$. The hope is that the new integral $\int v\,du$ is simpler than the original $\int u\,dv$. If it isn't, you chose the wrong parts!
2. Choosing the Parts — The LIATE Guideline
The strategic question is: which factor should be $u$ (differentiated) and which should be $dv$ (antidifferentiated)?
The LIATE Priority
A useful heuristic — not an unbreakable law — ranks function types by how much they want to be $u$ (differentiated):