Section 5 7 Antiderivatives Definition Function F x is called an antiderivative of f x on an interval I if F x f x for all x I Theorem 1 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 a constant Table of antidifferentiation formulas Function Antiderivative af x a is a constant aF x C f x g x F x G x C a a is a constant ax C x xn n 6 1 1 x ex ax sin x x2 C 2 xn 1 C n 1 ln x C ex C ax C ln a cos x C cos x sin x C sec2 x tan x C csc2 x cot x C 1 1 x2 1 1 x2 sin 1 x C tan 1 C 1 Example 1 Find the most general antiderivative of the function a f x x3 4x2 17 b f t sin t t 3 c f x 1 x2 x2 d f x x2 x 1 x e f x xe 1 5 1 x2 1 x2 Example 2 Find f x if 1 a f x 3 x f 1 2 x 2 b f x x f 0 3 f 0 2 The geometry of antiderivatives Example 3 Given the graph of a function f x Make a rough sketch of of an antiderivative of F given that F 0 0 Example 4 If f x 1 x4 1 sketch the graph of those antiderivatives F that satisfy the initial conditions F 1 1 F 0 0 F 1 1 3 3 2 2 y x y x 1 K 3 K 2 K 0 1 1 1 2 K 3 3 x K 2 K 0 1 K K K K K 1 2 2 3 3 3 2 x K 1 1 3 Rectilinear motion If the object has a position function s s t then v 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 functions Definition A vector function R t X t Y t is called an antiderivative 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 is an antiderivative of r on an interval I then the most general antiderivative Theorem 2 If R of r on I is C R is an arbitrary constant vector where C Example 6 Find the vector function that describe the position of particle that has an acceleration a t cos t et and v 0 r 0 0 4
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