4.1 Antiderivatives and Indefinite Integration

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) = –sinx, then the derivative of F(x) = cosx + 4 is the given function f(x).

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 then F1(x) = sinx – cosx – 1 F2(x) = sinx – cosx + 4 F3(x) = sinx – cosx – 9 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.

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

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

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) = sinx + cosx

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

Notation of Antiderivatives Variable of Integration Constant of Integration Integrand single function bag of functions

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

Example Example 2 Find the antiderivatives of 3 Solution constant

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

Example

Basic Integration Rules

Example Example 4 Find the antiderivatives of the following and Solution

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

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

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

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

Practice

Practice Check:

Practice check:

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

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

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

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

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

USED MOST OFTEN

USED MOST OFTEN

Example Example 9 Solut1ion Check:

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

Motion Along a Line s(t) position function v(t) = s’ (t) v(t) velocity a(t) = v’ (t) a(t) acceleration v(t) = s’ (t) a(t) = v’ (t) a(t) = s’’ (t)

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

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

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

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

Homework Pg. 255 9-13 odd, 21-33 odd not 27, 47, 48, 49, 51, 67, 69, 75, 81