Don't integrate - balloontegrate!
Straight lines integrate up, differentiate down (making it easier to check). The triangular networks satisfy the product rule - you may prefer to draw a zoomed-out version of these...
The challenge of integration by parts is simply to fill out a product rule shape starting with one of the bottom balloons instead of (as happens in differentiation) the top one.
See Balloon Calculus: worked examples from past papers
to see the process.