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Most calculus students might think that if one could compute indefinite integrals, it would always be easy to compute definite ones. After all, they might think, the Fundamental Theorem of Calculus says that one just has to subtract the values of the indefinite integral at the end points to get the definite integral.
So how come inside Mathematica there are thousands of pages of code devoted to working out definite integrals—beyond just subtracting indefinite ones?
Mathematica and the Fundamental Theorem of Calculus