How do integral calculators find the antiderivative of a function?
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If they do find it, they do it through symbolic algebra, the same way you do. But if you’re talking about your TI-83 or something, it doesn’t. It just basically does a Riemann sum, adding up a bunch of thin rectangles under the curve of the function and giving you an approximate value for the integral.
Symbolic calculator (the one that compute antiderivative exactly and can manipulate mathematical expressions) use something like Risch’s algorithm. The details are pretty complicated, but in simple term, it’s a much more advanced version of partial fraction decomposition by making use of Louisville’s algorithm (which describe the form of the solutions) to figure out what substitution to do.
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