# The Indefinite Integral

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The Indefinite Integral
Objective: Develop some fundamental results about antidifferentiation

Definition Definition 5.2.1
A function F is called an antiderivative of a function f on a given interval I if F/(x) = f(x) for all x in the interval.

Definition Definition 5.2.1
A function F is called an antiderivative of a function f on a given interval I if F/(x) = f(x) for all x in the interval. For example, the function is an antiderivative of on the interval because for each x in this interval

Definition However, is not the only antiderivative of f on this interval. If we add any constant C to this would also be an antiderivative. We will express this as

Theorem 5.2.2 If F(x) is any antiderivative of f(x) on an interval I, then for any constant C the function F(x) + C is also an antiderivative on that interval. Moreover, each antiderivative of f(x) on the interval I can be expressed in the form F(x) + C by choosing the constant C appropriately.

The Indefinite Integral
The process of finding antiderivatives is called antidifferentiation or integration. Thus, if then integrating the function f(x) produces an antiderivative of the form F(x) + C. This is written

The Indefinite Integral
The process of finding antiderivatives is called antidifferentiation or integration. Thus, if then integrating the function f(x) produces an antiderivative of the form F(x) + C. This is written Note that if we differentiate an antiderivative of f(x), we obtain f(x) again. Thus

The Indefinite Integral
The differential symbol dx, in the differentiation and antidifferentiation operations serves to identify the independent variable. If an independent variable other than x is used, then we would change the notation appropriately.

The Power Rule in Reverse
Lets look at the Power Rule again. We are going to do everything in reverse. Differentiation Integration Mult. by the exponent Add 1 to the exponent Subt. 1 from exponent Div by the new exponent

The Power Rule in Reverse
Lets look at the Power Rule again. We are going to do everything in reverse. Differentiation Integration Mult. by the exponent Add 1 to the exponent Subt. 1 from exponent Div by the new exponent

The Power Rule in Reverse
Here are some more examples. This process needs to be memorized.

Integration formulas Here are some examples of derivative formulas and their equivalent integration formulas. These need to be memorized. You can find them on page 357.

Integration formulas Here are some examples of derivative formulas and their equivalent integration formulas. These need to be memorized. You can find them on page 357.

Properties of the Indefinite Integral
Theorem 5.2.3 Suppose that F(x) and G(x) are antiderivatives of f(x) and g(x) respectively, and that c is a constant. Then: A constant factor can be moved through an integral sign.

Properties of the Indefinite Integral
Theorem 5.2.3 Suppose that F(x) and G(x) are antiderivatives of f(x) and g(x) respectively, and that c is a constant. Then: b) An antiderivative of a sum is the sum of the antiderivative.

Properties of the Indefinite Integral
Theorem 5.2.3 Suppose that F(x) and G(x) are antiderivatives of f(x) and g(x) respectively, and that c is a constant. Then: b) An antiderivative of a diff. is the diff of the antiderivative.

Properties of the Indefinite Integral
Theorem can be summarized by the following formulas.

Example 2 Evaluate

Example 2 Evaluate

Example 2 Evaluate

Example 2 Evaluate

Example 3 Evaluate

Example 3 Evaluate

Example 4 Evaluate

Example 4 Evaluate

Example 4 Evaluate

Example 4 Evaluate

Example 4 Evaluate

Example 4 Evaluate

Integral Curves Graphs of antiderivatives of a function are called integral curves of f. For example, is one integral curve for All other integral curves have equations of the form

Example 5 Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1).

Example 5 Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1). “The tangent line has slope x2” means that is the value of the derivative of the function. To find the function, we integrate.

Example 5 Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1). “The tangent line has slope x2” means that is the value of the derivative of the function. To find the function, we integrate.

Example 5 Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1). Now we use the point (2, 1) to solve for C.

Example 5 Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that each point (x, y) on the curve, the tangent line has slope x2. Find an equation for the curve given that it passes through the point (2, 1). Now we use the point (2, 1) to solve for C.

Homework Section 5.2 1-35 odd