Section 4.1 – Antiderivatives and Indefinite Integration.

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Section 4.1 – Antiderivatives and Indefinite Integration

Reversing Differentiation We have seen how to use derivatives to solve various contextual problems. For instance, if the position of a particle is known, then both the velocity and acceleration can be calculated by taking a derivative: But what if ONLY the acceleration of a particle is known? It would be useful to determine its velocity or its position at a particular time. For this case, a derivative is given and the problem is that of finding the corresponding function. Position FunctionThe derivative of the Position Function is the Velocity Function The derivative of the Velocity Function is the Acceleration Function Acceleration Function What function has a derivative of 32? What function has a second derivative of 32?

Antiderivative A function F is called an antiderivative of a given function f on an interval I if: for all x in I. Example:

The Uniqueness of Antiderivatives Suppose, find an antiderivative of f. That is, find a function F ( x ) such that. Using the Power Rule in Reverse Is this the only function whose derivative is 3 x 2 ? There are infinite functions whose derivative is 3 x 2 whose general form is: C is a constant real number (parameter)

Antiderivatives of the Same Function Differ by a Constant If F is an antiderivative of the continuous function f, then any other antiderivative, G, of f must have the form: In other words, two antiderivatives of the same function differ by a constant.

Differential Equation A differential equation is any equation that contains derivatives. If a question asks you to “solve a differential equation,” you need to find the original equation (most answers will be in the form y=). Example: The following is a differential equation because it contains the derivative of G : The general solution to the differential equation is:

Example 1 Find the general antiderivative for the given function. Using the opposite of the Power Rule, a first guess might be: But: Divide this result by 6 to get x 5 If:Then: General Solution:

Example 2 Find the general antiderivative for the given function. Using the opposite of the Trigonometric Derivatives, a first guess might be: But: Multiply this result by -1 to get sinx If:Then: General Solution:

Example 3 Find the general antiderivative for the given function. Using the opposite of the Power Rule, a first guess might be: But: Divide this result by 4 to get 5x 3 If:Then: General Solution:

Example 4 Find the general antiderivative for the given function. Using the opposite of the Power Rule, a first guess might be: But: Multiply this result by 2 to get x -1/2 If:Then: General Solution: Rewrite if necessary

Example 5 Find the general antiderivative for the given function. Using the opposite trigonometry derivatives: But: Multiply this result by 1/2 to get 9sec 2 2x If:Then: General Solution: Rewrite if necessary

Antiderivative Notation The notation Means that F is an antiderivative of f. It is called the indefinite integral of f and satisfies the condition that for all x in the domain of f. Integral Variable of Integration Constant of Integration Indefinite Integral

New Notation with old Examples Find each of the following indefinite integrals.

Basic Integration Rules Constant Multiple Sum Rule Difference Rule Constant Rule (zero) Let f and g be functions and x a variable; a, b, and c be constant; and C is an arbitrary constant.

Basic Integration Rules Constant Rule (non-zero) Power Rule Trigonometric Rule Let f and g be functions and x a variable; a, b, and c be constant; and C is an arbitrary constant.

Basic Integration Rules Trigonometric Rule Let f and g be functions and x a variable; a, b, and c be constant; and C is an arbitrary constant.

Example 1 Evaluate Sum and Difference Rules Constant Multiple Power and Constant Rules Simplify

Example 2 Evaluate Rewrite Sum Rule Constant Multiple Rule Simplify Power and Trig Rules

Example 3 The graph of a certain function F has slope at each point ( x, y ) and contains the point (1,2). Find the function F. Difference Rule Constant Multiple Rule Simplify Power and Constant Rules Integrate: Use the Initial Condition to find C:

Example 4 A particle moves along a coordinate axis in such a way that its acceleration is modeled by for time t>0. If the particle is at s=5 when t=1 and has velocity v=-2 at this time, where is it when t=4 ? Integrate the acceleration to find velocity: Use the Initial Condition to find C for velocity: Integrate the Velocity to find position: Use the Initial Condition to find C for position: Answer the Question: