4.1 The Indefinite Integral

Antiderivative An antiderivative of a function f is a function F such that Ex.An antiderivative of since is

means to find the set of all antiderivatives of f. The expression: read “the indefinite integral of f with respect to x,” Integral sign Integrand Indefinite Integral x is called the variable of integration

Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Notice Constant of Integration Represents every possible antiderivative of 6x.

Power Rule for the Indefinite Integral, Part I Ex.

Sum and Difference Rules Ex. Constant Multiple Rule Ex.

Position, Velocity, and Acceleration Derivative Form If s = s(t) is the position function of an object at time t, then Velocity = v =Acceleration = a = Integral Form

DIFFERENTIAL EQUATIONS A differential equation is an equation that contains a derivative. For example, this is a differential equation. From antidifferentiating skills from last chapter, we can solve this equation for y.

THE CONCEPT OF THE DIFFERENTIAL EQUATION The dy/dx = f(x) means that f(x) is a rate. To solve a differential equation means to solve for the general solution. By integrating. It is more involved than just integrating. Let’s look at an example:

EXAMPLE 1 GIVEN Multiply both sides by dx to isolate dy. Bring the dx with the x and dy with the y. Since you have the variable of integration attached, you are able to integrate both sides. Note: integral sign without limits means to merely find the antiderivative of that function Notice on the right, there is a C. Constant of integration.

C?? What is that? Remember from chapter 2? The derivative of a constant is 0. But when you integrate, you have to take into account that there is a possible constant involved. Theoretically, a differential equation has infinite solutions. To solve for C, you will receive an initial value problem which will give y(0) value. Then you can plug 0 in for x and the y(0) in for y. Continuing the previous problem, let’s say that y(0)=2.

Solving for c. Continuing the previous problem, let’s say that y(0)=2.

Basic Integration Rules