Reversing the Product Rule

Calculus 2 · Axiom Academy

A rocket's force fades while it speeds up. To get the work it does, you have to integrate a product — and that's exactly what integration by parts is for. A thruster pushes with a force that decays , F(t) = 10e^ -0.5t N, while the craft's velocity climbs , v(t) = 2t m/s. The work it does is — the integral of a product . Three moves turn that into an answer. Drag time forward. Force F (blue) drops as velocity v (red) rises, so their product (the power) swells then fades — and the shaded area beneath it is the work done so far , . It's the running total of a product. That product is what we have to integrate. Undo the product rule — pick u and dv The product rule says . Run it backwards and you get integration by parts: . The whole game is which factor you call u . The good choice makes the leftover integral simpler ; the bad one makes it worse . Flip the switch and watch. With the good choice, the formula does the rest. Step through it: the uv term, then the leftover , combine into the antiderivative. Land on a time and the bar matches the same shaded work you watched grow in Beat 1. One idea, three moves: feel that work is the integral of a product, reverse the product rule by choosing u so the leftover integral simplifies, then assemble . Any time a product of functions needs integrating — a polynomial times e^x , , — integration by parts is the product rule run backwards.

This is the written version of the interactive lesson above. See the full Calculus 2 course.