4.1a Antiderivatives and 4.1a Antiderivatives and Indefinite IntegrationIndefinite Integration

Write the general solution of a differential equationUse indefinite integration for antiderivativesUse basic integration rules to find antiderivativesFind a particular solution of a differential equation

Suppose you were asked to find the function F(x) whose derivative is f(x) = 3x2.

What would you come up with?

How about if the derivative was f(x) = x2?

How about if the derivative was f(x) = 2x4?

Notice that F(x) is called an antiderivative, not the antiderivative. See why:F1(x) = x3 F2(x) = x3 + 2 F3(x) = x3 – 5

Each would have the same derivative, f(x) = 3x2

In other words, any constant added on would give same results.

You get a whole FAMILY of antiderivatives by adding a constant C to the known antiderivative. A point on the curve of the antiderivative might be needed to nail down what the constant is for a specific case. C is called the constant of integration.

Knowing that Dx[x2] = 2x, all antiderivatives of f(x) = 2x would be represented by G(x) = x2 + C ; this would be called the general antiderivative, and G(x) is the general solution of the differential equation G’(x) = 2x

A differential equation in x and/or y is an equation that involves x, y, and the derivatives of y. y‘ = 3x and y’ = x2 +1 are examples of differential equations.

Ex 1 p. 249 Solving a differential equation

Find the general solution to the differential equation y’ = 3

Solution: You need to find a function whose derivative is 3. How about y = 3x? With info from theorem 4.1, the general solution could be y = 3x + C. 4

2

-2

-4

C = 3

C = 0

C = -2

Notation for antiderivatives:When solving a differential equation of the form

It is easier to write in the equivalent dy = f(x)dx form. The operation of finding all solutions of differential equations is called antidifferentiation or indefinite integration.Notation:

(read antiderivative of f with respect to x. Notice that dx tells you what variable you are integrating with respect to.

dy

dxf x ( )

( ) ( )y f x dx F x C

Integration is the “inverse” of differentiation

Differentiation is the “inverse”of integration

'( ) ( )F x dx F x C

( ) ( )dy

f x dx f xdx

These will need to be memorized now too!

Ex 2 Applying the basic integration rules

Find the antiderivatives of 5x

Solution:

Since C represents ANY constant, we could write in the simpler form 5

22x C

5xdx

225

5 5( ) 52 2

xxdx C x C

Ex. 3 p. 251 Rewriting before Integrating

a. 1

3x3 C

b. 3

4

43x C

c. 3cos x C

4

1a. dx

x

3b. xdx

c. 3sin x dx

Ex 4 p 252 Integrating Polynomial Functions

x C

x

C x C2

1 225

x

x C2

25

FHG

IKJ

FHG

IKJ 2

65

23

6 2x xx C

1

3

5

236 2x x x C

. a dx 1 dx. ( 5)b x dx 5 x dx dx

5. (2 5 3)c x x dx

Ex 5 p. 252 Rewriting before Integrating

2

32 15x x Cb g

2 5xdx

x

Ex 6 p252 Rewrite before Integrating

csc x C

2

cos

sin

xdxx

One of the most important steps to integration is REWRITING the integrand in a form that fits the basic integration rules.

4.1a p. 255/ 3-33 mult of 3, 35-45 odd