- 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=k∫f(x)dx
- Power Rule for Integrals:
- ∫xndx=xn+1n+1+C, for n≠−1
- 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 ∫(4x3−5x2+x−7)dx
Evaluate the indefinite integral: ∫(3x4−2sinx+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?