5.3 The Indefinite Integral

  1. 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
  2. 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)
  3. 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)
    • sinx
    • d/dx(-cosx) = sinx
    • x2
    • d/dx(1/3x3) = x2
  4. 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)
  5. 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
  6. 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
  7. 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
  8. 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 _________
    • integration
    • curves
    • antiderivatives
  9. 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
  10. The terms "antiderivative" and "indefinite integral" are used ________. In some textbooks, and antiderivative is called a _______ function
    • interchangeably
    • primitive function
  11. 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 
    • differntial
    • independent variable 
    • integration
  12. 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
  13. State The Power Rule for Integrals Theorem
    • ∫ xndx = xn+1/n+1 + C 
    • for n ≠ -1
  14. 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
  15. 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
  16. 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
  17. State the Linearity of the Indefinite Integral Theorem
    • Sum Rule: ∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx
    • Multiples Rule: ∫cf(x)dx = c∫f(x)dx
  18. 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
  19. 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
  20. We can think of an antiderivative as a solution to the different equation:
    dy/dx = ____
    dy/dx = f(x)
  21. Differential equation (Define)
    Differential equation: an equation relating an unknown function and its derivatives
  22. 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
  23. 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
  24. s'(t) = _____ and s(t) = ______
    v'(t) = _____ and v(t) = ______
    • s'(t) = v(t) and s(t) = ∫v(t)dt
    • v'(t) = a(t) and v(t) = ∫a(t)dt
Author
chikeokjr
ID
341863
Card Set
5.3 The Indefinite Integral
Description
5.3
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