antiderivatives and indefinite integration antiderivatives indefinite... · integration constant of...
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ANTIDERIVATIVES AND INDEFINITE INTEGRATION
Definition: A function F is an antiderivative of f on an interval I if for all x in I.
EX #1: Antiderivatives differ only by a constant, C:C is called the constant of integration
Family of all antiderivatives of f(x) = 2xand the general solution of the differential equation
The opposite of a derivative is called an antiderivative or integral.
A differential equation in x and y is an equation that involves x, y, and derivatives of y. For example:
and
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EX #3: Notation for antiderivatives:
EX #2: Solving a Differential Equation
differential form
Variable of Integration
Constant of Integration
Integrand
read as the antiderivative of f with respect to x.
So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative.
General Solution is denoted by:
The operation of finding all solutions of this equation is called antidifferentiation or indefinite integration denoted by sign.
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BASIC INTEGRATION RULES Integration is the “inverse” of differentiation.
Differentiation is the “inverse” of integration.
Differentiation Formula Integration Formula
POWER RULE:
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EX #4: Applying Basic Rules
EX #5: Rewriting Before Integrating
Original Integral Rewrite Integrate Simplify
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EX # 6: Polynomial Functions
A.
B.
C.
EX #7: Integrate By Rewriting
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EX #8: Solve differential equations subject to given conditions.
Given: and
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EX #9: A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet.
A. Find the position function giving the height, s, as a function of the time t.
B. When does the ball hit the ground?
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EX. #10: An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by dh/dt = 1.5t + 5, where t is the time in years, and h is the height in centimeters. The seedlings are 12 centimeters tall when planted (t = 0).
A. Find the height after t years. [Hint: the derivative is a rate of change of a function and the integral is the initial function.]
B. How tall are the shrubs when they are sold?