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4.1 The Indefinite Integral

4.1 The Indefinite Integral. Antiderivative. An antiderivative of a function f is a function F such that. Ex. An antiderivative of. is. since. Indefinite Integral. The expression:. read “the indefinite integral of f with respect to x ,”.

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4.1 The Indefinite Integral

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  1. 4.1 The Indefinite Integral

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

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

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

  5. Power Rule for the Indefinite Integral, Part I Ex.

  6. Sum and Difference Rules Ex. Constant Multiple Rule Ex.

  7. 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

  8. 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.

  9. 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:

  10. 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.

  11. 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.

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

  13. Basic Integration Rules

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