# 4: Evaluating Integrals

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• 4.1: Construction Accurate Graphs of Antiderivatives
Given the graph of a function f, we can construct the graph of its antiderivative F provided that (a) we know a starting value of F, say F(a), and (b) we can evaluate the integral R b a f (x) dx exactly for relevant choices of a and b. Thus, any function with at least one antiderivative in fact has infinitely many, and the graphs of any two antiderivatives will differ only by a vertical translation.
• 4.2: The Second Fundamental Theorem of Calculus
The Second Fundamental Theorem of Calculus is the formal, more general statement of the preceding fact: if f is a continuous function and c is any constant, then A(x) = R x c f (t) dt is the unique antiderivative of f that satisfies A(c) = 0. Together, the First and Second FTC enable us to formally see how differentiation and integration are almost inverse processes through the observations that Z x c d dt [ f (t)] dt = f (x) − f (c) and d dx "Z x c f (t) dt# = f (x).
• 4.3: Integration by Substitution
The technique of u-substitution helps us evaluate indefinite integrals of the form f (g(x))g' (x) dx through the substitutions u = g(x) and du = g' (x) dx. A key part of choosing the expression in x to be represented by u is the identification of a function-derivative pair. To do so, we often look for an “inner” function g(x) that is part of a composite function, while investigating whether g' (x) (or a constant multiple of g' (x)) is present as a multiplying factor of the integrand.

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