MATH 260 CHAPTER 25 - CHAPTER 25 Integration Section 1...

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CHAPTER 25: Integration Section 1: Antiderivatives 1. Determine the value of a that makes F(x) an antiderivative of f(x). a) An intiderivative of the function f(x) is a function F(x) such that F ‘(x). First, find F ‘(x). b) Now set F ‘(x) equal to f(x) and solve for a. F ‘(x) = f(x) 2. Determine the value of a that makes F(x) an antiderivative of f(x). , a) An antideriavative of the function f(x) is a function F(x) such that F ‘(x) = f(x). First, find F ‘(x). b) Now set F ‘(x) equal to f(x) and solve for a. F ‘(x) = f(x) = c) Now find a, multiply both side of the equation by the reciprocal of 3. Find the antiderivative of a) An antiderivative of the function f(x) is a function F(x) such that F ‘(x) = f(x). b) When finding the derivative of a constant times a power of x, one multiplies the constant coefficient by the power of x and reduces the power by 1. Therefore, working backwards, one must increase the power by 1. In this case, the power of x in the antiderivative must be c) Let and find its derivative d) Now equate F ‘(x) to f(x) and solve for a. F ‘(x) = f(x) e) Now substitute a = 1 into 4. Find an antiderivative of the function a) An antiderivative of the function f(x) is a function F(x) such that F ‘(x) = f(x). b) When finding the derivative of a constant times a power of x, one multiplies the constant coefficient by the power of x and reduces the power by 1. Therefore, working backwards, one must increase the power by 1. c) In this case, the power of x in the function f(x) is -9. Add 1 to determine the power of x in the antiderivative F(x).
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-9 + 1 = -8 d) Let and find its derivative e) Now set F ‘(x) equal to f(x) and solve for a. F ‘(x) = f(x) 8a = -7 f) Now substitute into 5. Find an antiderivative of the function . a) An antiderivative of the function f(x) is a function F(x) such that F ‘(x) = f(x) b) Notice the power of in the derivative, which implies that the antiderivative also has a power of . Since, in finding a derivative, 1 is subtracted from the power of , one should add 1 in finding the antiderivative. Thus, should be part of the antiderivative. c) Let F(x) = and find F ‘(x) d) Set F ‘(x) equal to f(x) and solve for a. F ‘(x) = f(x) e) Substitute into the equation for F(x). F(x) = 6. Find an antiderivative of the function a) An antiderivative of the function f(x) is a function F(x) such that F ‘(x) = f(x) b) Notice the power of 12x + 7 in the derivative, which implies that the antiderivative also has a power of 12x + 7. Since, in finding a derivative, 1 is subtracted from the power of 12x + 7, one should add 1 in finding the antiderivative. Thus, should be part of the antiderivative. c) Let F(x) = and find F ‘(x). d) Let F ‘(x) = f(x) and solve for a. F ‘(x) = f(x) e) Substitute into the equation for F(x) Section 2: The Indefinite Integral 1. Determine the following. a) The constant-multiple rule of differentiation states that a constant multiple k may be moved through the integral sign and is given by the following formula.
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b) Write according to the constant-multiple rule.
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