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 Function The 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 3x2? There are infinite functions whose derivative is 3x2 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:

Using the opposite of the Power Rule, a first guess might be: Example 1 Find the general antiderivative for the given function. Divide this result by 6 to get x5 Using the opposite of the Power Rule, a first guess might be: But: If: Then: General Solution:

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

Using the opposite of the Power Rule, a first guess might be: Example 3 Find the general antiderivative for the given function. Divide this result by 4 to get 5x3 Using the opposite of the Power Rule, a first guess might be: But: If: Then: General Solution:

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

Using the opposite trigonometry derivatives: Example 5 Find the general antiderivative for the given function. Rewrite if necessary Multiply this result by 1/2 to get 9sec22x Using the opposite trigonometry derivatives: But: If: Then: General Solution:

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. Indefinite Integral Constant of Integration Integral Variable of Integration

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

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

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

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

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

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

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 Integrate: Constant Multiple Rule Power and Constant Rules Simplify 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: