Namaste sathi! Hope everyone is doing well. I was revising Class 12 Calculus earlier today and got stuck thinking about how we often just memorize formulas like $\int u \, dv = uv - \int v \, udu$ without really visualizing what is happening under the hood.

It actually clicked for me when I linked it back to the standard Product Rule of differentiation. If you take $d(uv) = u \, dv + v \, du$ and just integrate both sides, the whole concept falls into place. You are basically taking the total area of a expanding rectangle and subtracting the piece you don’t need to find the area of the other piece. Once you see it as just the product rule in reverse, you don’t even need to rely purely on memorizing the ILATE rule mechanically.

How do you guys approach integration when a tricky question pops up? Do you prefer visual proofs like this, or do you rely on pattern recognition and practice problems?