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Antiderivatives and Indefinite Integrals

Definition: A function F is called an antiderivative of f on an interval I if F’(x) = f(x) for all x in I. Example: Let f(x)=x3. If F(x) =1/4 * x4 then F’(x) = f(x) Theorem: If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C where C is an arbitrary constant. F is an antiderivative of f

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Table of particular antiderivatives

Function Antiderivative xr, r -1 xr+1/(r+1) 1/x ln |x| sin x – cos x ex cos x ax ax / ln a sec2 x tan x tan-1 x sin-1 x af + bg aF + bG In the last and highlighted formula above we assume that F’ = f and G’ = g. The coefficients a and b are numbers. These rules give particular antiderivatives of the listed functions. A general antiderivative can be obtained by adding a constant .

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Computing Antiderivatives

Problem Solution Use here the formula (aF + bG)’ = aF’ + bG’

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Indefinite Integral Indefinite integral is a traditional notation for antiderivatives Note: Distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions). The connection between them is given by the Evaluation Theorem:

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Some indefinite integrals

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Analyzing the motion of an object using antiderivatives

Problem A particle is moving with the given data. Find the position of the particle. a(t) = t -3t2, v(0) = 2, s(0) = 5 v(t) is the antiderivative of a(t): v’(t) = a(t) = t -3t2 Antidifferentiation gives v(t) = 10t + 1.5t2 – t3 + C v(0) = 2 implies that C=2; thus, v(t) = 10t + 1.5t2 – t3 + 2 s(t) is the antiderivative of v(t): s’(t) = v(t) = 10t + 1.5t2 – t3 + 2 Antidifferentiation gives s(t) = 5t t3 – 0.25t4 + 2t + D s(0) = 5 implies that D=5; thus, s(t) = 5t t3 – 0.25t4 + 2t + 5 Solution

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