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6A (This accompanies part of section 7.2 of the book): Antidifferentiation Example 1: What functions have the following derivatives? (a) 3𝑥 2 (c) 15 (b) 4𝑥 3 (d) 1 𝑥 (e) sec2x (f) 1 √1−𝑥 2 Antidifferentiation: the process of finding a function 𝑓 whose known derivative is 𝑓 ′ (𝑥). Definition: Indefinite Integral The family of all antiderivatives of a function 𝑓(𝑥) is the indefinite integral of 𝒇 with respect to x and is denoted: ∫ 𝑓(𝑥)𝑑𝑥 If 𝐹 is any function such that 𝐹 ′ (𝑥) = 𝑓(𝑥), then ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶 where 𝐶 is an arbitrary constant called the constant of integration. ** the symbol ∫ is called the integral sign; f is called the integrand, x is the variable of integration Properties of Indefinite Integrals: 1. ∫ 𝑘𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥 2. ∫( 𝑓(𝑥) ± 𝑔(𝑥) )𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 ± ∫ 𝑔(𝑥)𝑑𝑥 Example 2: Find the most general antiderivative of the function: 1 𝑓(𝑥) = 𝑥 + 𝑒 𝑥 − cos 𝑥 Power Formula: ∫ 𝑢𝑛 𝑑𝑢 = 𝑢𝑛+1 𝑛+1 +𝐶 Example 2: Find the most general antiderivative for the function. Check you answer by differentiating: 𝑔(𝑥) = 𝑥 5 − 4𝑥 −3 + √𝑥 + 20 – csc 𝑥 cot 𝑥 **The constant of integration 𝐶 can be found if certain initial conditions exist as shown in the next example. Example 3: Find the antiderivative 𝐹 that satisfies the given conditions: 𝑓(𝑥) = 3𝑥 − sin 𝑥 and 𝑓(0) = 3 Example 4: Find 𝑓. 𝑓 ′′ (𝑥) = 20𝑥 3 − 10 𝑓(1) = 1, 𝑓 ′ (1) = −5