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Antiderivatives: Fresh Take

  • Understand indefinite integrals and learn how to find basic antiderivatives for functions
  • Use the rule for integrating functions raised to a power
  • Use antidifferentiation to solve simple initial-value problems

Finding the Antiderivative

The Main Idea 

  • Antidifferentiation is the reverse process of differentiation
  • A function F(x) is an antiderivative of f(x) if F(x)=f(x) for all x in the domain of f
  • General Form of Antiderivatives:
    • If F(x) is an antiderivative of f(x), then F(x)+C is the general form of all antiderivatives of f(x), where C is any constant

Find all antiderivatives of f(x)=sinx.

Find the general antiderivative of f(x)=xsin(x)+x2cos(x).

Indefinite Integrals

The Main Idea 

  • Definition of Indefinite Integral:
    • Notation: f(x)dx=F(x)+C
    • F(x) is an antiderivative of f(x)
    • C is the constant of integration
  • Terminology:
    • is the integral sign
    • f(x) is the integrand
    • x is the variable of integration
  • Key Properties:
    • Sum/Difference Rule: (f(x)±g(x))dx=f(x)dx±g(x)dx
    • Constant Multiple Rule: kf(x)dx=kf(x)dx
  • Power Rule for Integrals:
    • xndx=xn+1n+1+C, for n1
  • Common Indefinite Integrals:
    • 1xdx=ln|x|+C
    • exdx=ex+C
    • sinxdx=cosx+C
    • cosxdx=sinx+C
    • sec2xdx=tanx+C

Verify that xcosxdx=xsinx+cosx+C.

Evaluate (4x35x2+x7)dx

Evaluate the indefinite integral: (3x42sinx+5ex+4x)dx

Initial-Value Problems

The Main Idea 

  • Definition of Differential Equation:
    • An equation relating an unknown function and one or more of its derivatives
  • Initial-Value Problem:
    • A differential equation with an additional condition
    • Condition typically specifies the function value at a particular point
    • Example:
      • dydx=f(x),y(x0)=y0
  • Solving Initial-Value Problems:
    • Find the general solution of the differential equation
    • Use the initial condition to determine the specific solution
  • Applications:
    • Motion problems (position, velocity, acceleration)
    • Growth and decay models
    • Numerous real-world scenarios in physics, engineering, and other fields

Suppose the car is traveling at the rate of 44 ft/sec. How long does it take for the car to stop? How far will the car travel?