# 5.3 The Indefinite Integral

 .remove_background_ad { border: 1px solid #555555; padding: .75em; margin: .75em; background-color: #e7e7e7; } .rmbg_image { max-height: 80px; } In physics we may know the velocity v(t) (the derivative) and wish to compute the position s(t) of an object. Since s'(t) = v(t), this amounts to finding a function whose derivative is v(t). A function F whose derivative is f is called an ________ of f. ________ will turn out to be the key to evaluating definite integrals antiderivative Antiderivative Define Antiderivative A function F is an antiderivative of f on an open interval (a,b) if F'(x) = f(x) for all x in (a,b) F(x) = -cosx is an antiderivative of f(x) = ______ because for all values of x, F'(x) = ________ = ______ = f(x) F(x) = 1/3x3 is an antideriative of f(x) = ______ because for all values of x,  F'(x) = _______ = ______ = f(x) sinxd/dx(-cosx) = sinxx2d/dx(1/3x3) = x2 One critical observation is that antiderivatives are not unique. We are free to add a constant C because the derivative of a constant is _____, and so, if F'(x) = _____, then (F(x) + C' =____. For example, each of the following is a antiderivative of x2: 1/3x3, 1/3x3 + 5, 1/3x3 - 6 zero f(x)f(x) Are there any antiderivatives of f other than those obtained by adding a constant to a given antiderivative F? No if f is defined on an open interval (a,b)**by The General Antiderivative Theorem State the General Antiderivative Theorem Let y = F(x) be an antiderivative of y = f(x) on (a,b). Then every other antiderivative on (a,b) is of the form y = F(x) + C for some constant C Find two antiderivatives of f(x) = cosx. Then determine the general antiderivative The functions F(x) = sinx and G(x) = sinx + 2 are both antiderivatives of f(x) = cosx. The general antiderivative is F(x) = sinx + C, where C is ay constant The process of finding an antiderivative is called ______. We will see why in the next section, when we discuss the connection between antiderivatives and areas under ______ given by the Fundamental Theorem of Calculus. Anticipating this result, we begin using the integral sign ∫, the standard notation for _________ integrationcurvesantiderivatives Indefinite Integral: The notation ∫ f(x)dx = F(x) + C means that F'(x) = ____ We say that y = F(x) + C is the general _______ or ______ ______ of y = f(x) f(x)antiderivative or indefinite integral The terms "antiderivative" and "indefinite integral" are used ________. In some textbooks, and antiderivative is called a _______ function interchangeablyprimitive function Define Integrand The symbol dx is a _______, it is a part of o=the integral notation and serves to indicate the _______ _______. The constant C is called the constant of _________. Integrand: The expression f(x) appearing in the integral sign differntialindependent variable integration Some indefinite integrals can be evaluated by ________ the familiar derivative formulas. For example, we obtain the indefinite integral of y = xn by reversing the Power Rule for derivatives reversing State The Power Rule for Integrals Theorem ∫ xndx = xn+1/n+1 + C for n ≠ -1 Prove the Power Rule for Integrals Theorem We just need to verify that F(x) = xn+1/n + 1 is an antiderivative of f(x) = xn F'(x) = d/dx(xn+1/n+1 + C)F'(x) = 1/n + 1((n+1)xn) = xn In words, the Power Rule for Integrals instructs to follow two steps in order to integrate a power of x. State the steps 1) add one to the exponent 2) divide by the new exponent The Power Rule is not valid for n = -1. In fact, for n = -1, what meaningless result do we obtain? It turns out that the antideriative of f(x) = x-1 is the ______ _______ ∫ x-1dx = (xn+1/n + 1) + C = x0/0 + C (meaningless)natural logarithm State the Linearity of the Indefinite Integral Theorem Sum Rule: ∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dxMultiples Rule: ∫cf(x)dx = c∫f(x)dx State the Basic Trigonometric Integrals ∫sinx dx = -cosx + C∫sec2x dx = tanx + C∫secx tanx dx = secx + C∫cosx dx = sinx + C∫csc2x dx = -cotx + C∫cscx cotx dx = -cscx + C Similarly, for any constant k ≠ 0, the formulas: 1) d/dx sin(kx) = k cos(kx) 2) d/dx cos(kx) = -ksin(kx) translate to the which indefinite integral formulas? ∫cos(kx)dx = 1/k sin(kx) + C∫sin(kx)dx = -1/k cos(kx) +C We can think of an antiderivative as a solution to the different equation: dy/dx = ____ dy/dx = f(x) Differential equation (Define) Differential equation: an equation relating an unknown function and its derivatives The unknown in Eq. (1) is a function y = F(x) whose derivative is _____; that is y = F(x) is an _________ of y = f(x). f(x)antiderivative Eq. (1) has infinitely many solutions (because the antiderivative is not unique), but we can specify a particular solution by imposing an ______ ______, that is by requiring that the solution satisfy y(x0) = y0 for some fixed values x0 and y0. A differential equation with an initial condition is called an ______ _____ _______ initial condition initial value problem s'(t) = _____ and s(t) = ______ v'(t) = _____ and v(t) = ______ s'(t) = v(t) and s(t) = ∫v(t)dtv'(t) = a(t) and v(t) = ∫a(t)dt .remove_background_ad { border: 1px solid #555555; padding: .75em; margin: .75em; background-color: #e7e7e7; } .rmbg_image { max-height: 80px; } Authorchikeokjr ID341863 Card Set5.3 The Indefinite Integral Description5.3 Updated2018-08-17T17:29:20Z Show Answers