Section 5.7 AntiderivativesDefinition. Function F (x) is called an antiderivative of f(x) on an interval I ifF′(x) = f(x)for all x ∈ I.Theorem 1. If F is an antiderivative of f on an interval I, then the most general antiderivativeof f on I isF (x) + Cwhere C is a constant.Table of antidifferentiation formulasFunction Antiderivativeaf(x), a is a constant aF (x) + Cf(x) + g(x) F (x) + G(x) + Ca, a is a constant ax + Cxx22+ Cxn, n 6= −1xn+1n + 1+ C1xln |x| + Cexex+ Caxaxln a+ Csin x −cos x + Ccos x sin x + Csec2x tan x + Ccsc2x −cot x + C1√1 − x2sin−1x + C11 + x2tan−1+C1Example 1. Find the most general antiderivative of the function.(a.) f(x) = x3− 4x2+ 17(b.) f(t) = sin t −√t(c.) f(x) = (1 + x2)3√x2(d.) f(x) =x2+ x + 1x(e.) f(x) = xe+51 + x2−1√1 − x2Example 2. Find f(x) if(a.) f′(x) = 3√x −1√x, f(1) = 22(b.) f′′(x) = x, f (0) = −3, f′(0) = 2The geometry of antiderivativesExample 3. Given the graph of a function f(x). Make a rough sketch of of an antiderivativeof F , given that F (0) = 0.Example 4. If f(x) = 1/(x4+ 1), sketch the graph of those antiderivatives F that satisfy theinitial conditions F (−1) = 1, F (0) = 0, F (1) = −1.xK3K2K101 2 3y(x)K3K2K1123xK3K2K101 2 3y(x)K3K2K11233Rectilinear motionIf the object has a position function s = s(t), thenv(t) = s′(t) (the position function is an antiderivative for the velocity function),a(t) = v′(t) (the velocity function is an antiderivative to the acceleration function)Example 5. A particle is moving with the acceleration a(t) = 3t + 8, s(0) = 1, v(0) = −2.Find the position of the particle.Antiderivatives of vector functionsDefinition. A vector function~R(t) =< X(t), Y (t) > is called an ant iderivative of ~r(t) =<x(t), y(t) > on an interval I if~R′(t) = ~r(t) that is, X′(t) = x(t) and Y′(t) = y(t).Theorem 2. If~R is an antiderivative of ~r on an interval I, then the most general antiderivativeof ~r on I is~R +~Cwhere~C is an arbitrary constant vector.Example 6. Find the vector-function that describe the position of particle that has an accel-eration ~a(t) = cos t~ı + et~ and ~v(0) = ~ı + ~, ~r(0)
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