Calculus×Computing
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Integration by parts

Level FundamentalDifficulty ★★★★★Method⌖ Open in the map

What is it?

The product rule read backwards: ∫u dv=uv−∫v du\int u\,\dd v = uv - \int v\,\dd u. It moves a derivative from one factor to the other — the trick behind the weak formulation of differential equations and finite elements.

Formulas

∫abu v′ dx=[u v]ab−∫abu′ v dx\int_a^b u\,v'\,\dd x = \big[u\,v\big]_a^b - \int_a^b u'\,v\,\dd x
∫xex dx=xex−ex+C\int x e^x\,\dd x = x e^x - e^x + C

Where it shows up in computing

  • Finite element method★★★★★fundamentalPhysics and simulation

    FEM rewrites −u′′=f-u'' = f as ∫u′v′=∫fv\int u'v' = \int fv by parts, lowering the smoothness required of uu.

  • Symbolic computation (CAS)★★★★★frequentScientific computing and algorithms

    A standard heuristic of symbolic integrators (LIATE ordering).

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

λ Scientific computing and algorithms

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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