4.1 Antiderivatives and Indefinite Integration. Suppose you were asked to find a function F whose...
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Transcript of 4.1 Antiderivatives and Indefinite Integration. Suppose you were asked to find a function F whose...
4.1 Antiderivatives and Indefinite Integration
23)( xxf
3)( xxF
Suppose you were asked to find a function F whose derivative is
From your knowledge of derivatives, you would probably say
The function F is an antiderivative of f. In general, a function F is an antiderivative of f (x) if
xxfxF )()('
Note that F is an antiderivative, not the antiderivative
Ex:
CxF )(
)(xf
In general:
is the antiderivative of
Example 1
xxf 2)( Example 2: Solving a Differential Equation
Gives the entire family of
antiderivatives
2'y
)(xfdx
dy
dxxfdy )(
Notation for Antiderivatives:When solving a differential
equation
it is convenient to write the differential form
CxFdxxfy )()(
The operation of finding all solutions of this equation is antidifferentiation or indefinite integration
Integrand
Variable of integration
Constant of integration
Practice
dxx3
1 dtt22 1
xdx3 dxx
Practice
dxxxx 24 53
dxx
x2cos
sin
dxx
x 1
xdxsin2
Initial Conditions and Particular Solutions
Solve the differential equation:
13 2 xdx
dy
Solve the differential equation above if the curve passes through (2,4)—called an initial condition.
Initial Conditions and Particular Solutions
Find the general solution of:
0,1
)('2
xx
xF
and find the particular solution that satisfies the initial condition F(1)=0
A Vertical Motion Problem
A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet, as shown in the figure.
1. Find the position function giving the height s as a function of time t2. When does the ball hit the ground?