1 5.1: Antiderivatives and Indefinite Integrals Introduction to Antiderivative If f’(x) = x², then f(x) = _________________? If F(x) represents the “original” function, then we say F(x) = ?³3is an antiderivativeof the function f(x) = x² because F’(x) = 𝑑𝑑?(?33)= x² However, there are manyfunctions that have a derivative of f(x) = x², such as: In summary, antidifferentiationof a given function results in ________________________. In our example, we would write: ______________________________________________. The Indefinite Integral ∫?(?)??∫ ?(?)??= F(x) + C if F’(x) = f(x) integral signintegrandconstant of integrationAntiderivative A function Fis an antiderivative of a function fif: 𝑭′(𝒙) = 𝒇(𝒙)Objectives ▪The student will be able to formulate problems involving antiderivatives. ▪The student will be able to use the formulas and properties of antiderivatives and indefinite integrals. ▪The student will be able to solve applications using antiderivatives and indefinite integrals.
2 Example: Find each indefinite integral a) ∫ 5??b) ∫ 9??