4_1_Antiderivative - 4.1 Antiderivatives: Basic Concepts A...

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Unformatted text preview: 4.1 Antiderivatives: Basic Concepts A Differential Equation in x and y is an equation that involves, x, y, and the derivative of y. eg: dy/dx = 5x4 6x2 Definition, Antiderivative: A function F is an antiderivative of f on an interval I if F ' (x) = f(x) for all x in I. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework on Web Title: Basic Concepts (1 of 5) 4.1 Antiderivatives: Basic Concepts Antidifferentiation or Indefinite Integration The operation of finding all the solutions to the differential equation, dy = f (x) dx, is called antidifferentiation or indefinite integration and is denoted by the integral sign . The solution when F '(x) = f(x) is: y = f(x)dx = F(x) + c integrand variable constant of integration of integration Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework on Web Title: Antidifferentiation (2 of 5) y = f(x)dx = F(x) + c integrand variable constant of integration of integration Note: Operators and dx are no longer present after the operation of integration is performed. The integral sign and dx indicate the operation of integration the same way that a plus sign indicates the operation of addition. For the addition problem, 3 + 2 = 5, the result no longer has the operator, + it contains only the result. Likewise, for integration, the result no longer has either the integral sign or the dx. To continue to write the or the dx after the operation of integration has been performed is, therefore, not correct. Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework on Web Title: Operator not present (3 of 5) Rules of: Differentiation Integration d( c ) = 0 dx d(kx) = k dx 0 dx = c k dx = kx + c d(xn = nxn-1 ) dx xndx = xn+1 c + (n+1) d(sinx) = cos x dx d(cosx) = -sin x dx | | | | | cosxdx = sinx + c sinxdx = -cosx + c | | | | | Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework on Web Title: Rules of Integration (4 of 5) Homework: 4.1 # 1, 7, 9, 13, 21, 25, 29, 33, 37, 41, 55, 59, 61, 63, 67 Class Notes: Prof. G. Battaly, Westchester Community College, NY Calculus Home Page Homework on Web Title: Homework (5 of 5) ...
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