Antiderivatives.mws

Applications of Differentiation: Antiderivatives

Vocabulary

antiderivative, general antiderivative, indefinite integral, Leibniz symbol, Power Rule for Integrals, Generalized Power Rule for Integrals, linearity

Objectives

Lecture Outline

Definition of Antiderivative

We say that F(x) is an antiderivative of f(x) on an interval I if Diff(F(x),x) = f(x) for all x in I.

Example: The functions F(x) = x^2 and G(x) = x^2+7 are both antiderivatives of f(x) = 2*x .

When we studied the Mean Value Theorem, we developed a theorem which shows that any two antiderivatives of a function (on an interval) must differ by a constant. Hence, if F(x) is an antiderivative for f(x) , then for any constant C , F(x)+C is also an antiderivative for f(x) .

Example: Each member F(x) of the family of functions sin(x)+C (where C is constant) is an antiderivative of f(x) = cos(x) .

Definition of Indefinite Integral

Int(f(x),x) = F(x)+C if and only if F(x) is an antiderivative for f(x) .

We say that the general antiderivative of f(x) is F(x)+C .

The process of antidifferentiation (that is, the process of finding an antiderivative for a function) is essentially the inverse of the process of differentiation. For each differentiation rule we have developed, there is a corresponding antidifferentiation rule.

Consider the Power Rule for differentiation. We have that Diff(x^n,x) = n*x^(n-1) for all real numbers n . The corresponding indefinite integral statement provides us with the Power Rule for Integrals.

Power Rule for Integrals

Int(x^n,x) = x^(n+1)/(n+1)+C , for all n not equal to -1.

Example: Int(x^7,x) = x^8/8+C .

Suppose that we extend the last example by considering the indefinite integral of a power of a general function f(x) . Note that by the Powerchain Rule for differentiation,

Diff(f(x)^(n+1)/(n+1),x) = f(x)^n*diff(f(x),x) .

The corresponding indefinite integral statement is the Generalized Power for Integrals.

Generalized Power Rule for Integrals

Int(f(x)^n*diff(f(x),x),x) = f(x)^(n+1)/(n+1)+C , for n not equal to -1.

Example: Int((x^3+1)^10*3*x^2,x) = (x^3+1)^11/11+C .

The indefinite integral is a linear operator, just as the limit and differentiation operators are also linear. This means that the indefinite integral of a constant multiple times a function is the constant times the indefinite integral of the function and the indefinite integral of a sum of functions is the sum of the indefinite integrals of the functions.

Example: Int(x^3+3,x) = Int(x^3,x)+Int(3,x) , which is x^4/4+3*x+C .