Antiderivatives

Think About It Suppose this is the graph of the derivative of a function What do we know about the original function? Critical numbers Where it is increasing, decreasing What do we not know? 2 f '(x)

The work to this point has involved finding and applying the first or second derivative of a function. In this chapter we will reverse the process. If we know the derivative of a function how do we obtain the original function? The process is called antidifferentiation or integration.

Anti-Derivatives Derivatives give us the rate of change of a function What if we know the rate of change … Can we find the original function? If F '(x) = f(x) Then F(x) is an antiderivative of f(x) Example – let F(x) = 12x 2 Then F '(x) = 24x = f(x) So F(x) = 12x 2 is the antiderivative of f(x) = 24x 4

Finding An Antiderivative Given f(x) = 12x 3 What is the antiderivative, F(x)? Use the power rule backwards Recall that for f(x) = x n … f '(x) = n x n – 1 That is … Multiply the expression by the exponent Decrease exponent by 1 Now do opposite (in opposite order) Increase exponent by 1 Divide expression by new exponent 5

Family of Antiderivatives Consider a family of parabolas f(x) = x 2 + n which differ only by value of n Note that f '(x) is the same for each version of f Now go the other way … The antiderivative of 2x must be different for each of the original functions So when we take an antiderivative We specify F(x) + C Where C is an arbitrary constant 6 This indicates that multiple antiderivatives could exist from one derivative

Indefinite Integral The family of antiderivatives of a function f indicated by The symbol is a stylized S to indicate summation 7

Indefinite Integral The indefinite integral is a family of functions The + C represents an arbitrary constant The constant of integration 8

Properties of Indefinite Integrals The power rule The integral of a sum (difference) is the sum (difference) of the integrals 9

Properties of Indefinite Integrals The derivative of the indefinite integral is the original function A constant can be factored out of the integral 10

Example : Evaluate

Example : Find the function f such that First find f (x) by integrating.

Example : Evaluate and check by differentiation:

Examples Determine the indefinite integrals as specified below 14

Integrate

Find each antiderivative

Solve the differential equation

Given that the graph of f(x) passes through the point (1,6) and that the slope of its tangent line at (x.f(x) is 2x+1, find f(6)