calc 4.1a
TRANSCRIPT
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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
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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?
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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.
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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.
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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
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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
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Integration is the “inverse” of differentiation
Differentiation is the “inverse”of integration
'( ) ( )F x dx F x C
( ) ( )dy
f x dx f xdx
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These will need to be memorized now too!
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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
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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
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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
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Ex 5 p. 252 Rewriting before Integrating
2
32 15x x Cb g
2 5xdx
x
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Ex 6 p252 Rewrite before Integrating
csc x C
2
cos
sin
xdxx
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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