Take the Fraction Apart First

You'll learnto check the two conditions a decomposition needs before writing a single letter, to solve for the constants by two routes that have to agree, to integrate each piece into a logarithm with its bars and its sign intact — and to test the whole decomposition by putting the pieces back and comparing, which takes fifteen seconds and catches every sign error.

A fraction whose denominator factors can be taken apart into simpler ones, each of which integrates to a logarithm. The method is algebra, not calculus, and it fails in algebraic ways: the wrong factor set, the wrong shape of numerator, or constants that were never solved for at all. The last of those is the expensive one, because a split with no constants in it looks like a decomposition and is a different function — which is a claim you can test by plotting both and looking, and that is what this file does.

8 STEPS · 6 QUICK CHECKS · PUT THE PIECES BACK AND COMPARE · v1

Factor, set one constant over each factor, solve, then integrate each piece

Before you start

What you're looking at

One plot panel. The blue curve is always the rational function being taken apart. The pink and violet curves are the two pieces of its decomposition — violet is the piece whose constant came out negative. A yellow dashed curve is a measured quantity: the two pieces added back together, or the measured slope of an antiderivative.

The question

What does a decomposition actually claim, and how do you know in seconds whether the pieces you wrote down are the right ones?

Watch for

A correct decomposition puts one curve on the screen, not two, because the pieces added back ARE the original function. When a second curve appears, something is wrong — and the card prints how far apart they get.

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