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4.1 Antiderivatives and Indefinite Integration

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Definition of an Antiderivative In many cases, we would like to know what is the function F ( x ) whose derivative is a given function f(x). Or, F ( x ) = f(x) For instance, if a given function f ( x ) = –sin x, then the derivative of F ( x ) = cos x + 4 is the given function f(x).

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Definition of an Antiderivative Notice that even we could find the antiderivative of a given function, the answer is not unique! If f(x) = sinx + cosx This is why F ( x ) is called an antiderivative of f(x), rather then the antiderivative of f(x). In fact, all antideritives of f(x) differs at a constant. then F 1 ( x ) = sinx – cosx – 1 F 2 ( x ) = sinx – cosx + 4 F 3 ( x ) = sinx – cosx – 9

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Antiderivatives and Indefinite Integration Differentiable Functions ALL functions D derivative operator antiderivative operator/ indefinite integral operator is read as the antiderivative of f with respect to x or the indefinite integral of f with respect to x. This expression denotes (all) the antiderivatives of f(x).

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Theorem 4.1 Representation of Antiderivatives This Theorem tells us that we can represent the entire family of antideritives of a given function by adding a constant to a known antiderivative. If f(x) = sinx + cosx then F ( x ) = sinx – cosx + C

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Theorem 4.1 Representation of Antiderivatives The constant C is called the constant of integration. The family of functions represented by G is the general antiderivative of f. If f(x) = sinx + cosx then F ( x ) = sinx – cosx + C is the general solution of the differential equation F ( x ) = sin x + cos x

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Notation of Antiderivatives Let F(x) be an antiderivative of f(x), and y be all antiderivatives f(x), then We differentiate (*), then (*) The operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration) denoted by the sign

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Notation of Antiderivatives single functionbag of functions Variable of Integration Integrand Constant of Integration

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single functionbag of functions iff F (x) = f(x) Example 1 Find the antiderivatives of 5 x Indefinite Integration Solution

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Example constant Example 2 Find the antiderivatives of 3 Solution

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Example 3 Find the antiderivatives of the following Solution and m is not necessary to be an integer. m ψ R

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Example

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Basic Integration Rules

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Example 4 Find the antiderivatives of the following Solution Example and

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Example 5 Find the antiderivatives of the following Solution Example Also refer to middle of P. 251 for another example.

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Example 6 Find the antiderivatives of the following Solut1ion Example Tip: Sometimes, simplifying or rewriting before integration

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(1) Please be aware that the indefinite integral Note is NOT equal to

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(2)The indefinite integration with respect to a polynomial Note can NOT be written as

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Practice

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Check:

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check: Practice

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Initial Conditions and Particular Solutions In the beginning of this section, we have already known that all antideritives of f(x) differs at a constant. This means that the graphs of any two antiderivatives of f are VERTICAL translation of each other. C=2 C=1 C=0 C=–1 C=–2

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Initial Conditions and Particular Solutions C=2 C=1 C=0 C=–1 C=–2 Graph must pass through point (0, –2). So, x = 0, y = –2 is the solution of the equation Or,

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(To find a particular solution, we need an initial condition) Example 7 Find f(x) if f ( x ) = 6 x – 4 and f (1) = 2. differential equation use this to get a bag of functions containing f(x) initial condition use this to reach into bag and pull out a particular f(x)

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Example 7 Find f(x) if f (x) = 6x – 4 and f(1) = 2. check: Solut1ion

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Example 8 Find f(x) if differential equationinitial condition Answer: Example Solut1ion

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USED MOST OFTEN

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Example 9 Check: Example Solut1ion

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Example Example 10 Find the antiderivatives of the following check: Solut1ion

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s(t)s(t) position function v(t)v(t) velocity a(t)a(t) acceleration v(t) = s (t) a(t) = v (t) v(t) = s (t) a(t) = v (t) a(t) = s (t) Motion Along a Line

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NOTE Velocity is negative when falling (or positive when thrown up) acceleration due to gravity Free Falling Objects

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Example Example 11 The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as function of the time t in the seconds. How long will it take the rock to hit the canyon floor? Solut1ion Let h(t), v(t) and a(t) be the height, the velocity, and the acceleration of the rock at time t. Then

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Example Solut1ion Let h(t), v(t) and a(t) be the height, the velocity, and the acceleration of the rock at time t. Then So

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Example Solut1ion When h(t) = 0, the rock will hit the canyon floor. So (The negative root will be discard)

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Homework Pg odd, odd not 27, 47, 48, 49, 51, 67, 69, 75, 81

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