Summary of Antiderivatives

Essential Concepts

  • If [latex]F[/latex] is an antiderivative of [latex]f[/latex], then every antiderivative of [latex]f[/latex] is of the form [latex]F(x)+C[/latex] for some constant [latex]C[/latex].
  • Solving the initial-value problem
    [latex]\frac{dy}{dx}=f(x),y(x_0)=y_0[/latex]

    requires us first to find the set of antiderivatives of [latex]f[/latex] and then to look for the particular antiderivative that also satisfies the initial condition.

Glossary

antiderivative
a function [latex]F[/latex] such that [latex]F^{\prime}(x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex] is an antiderivative of [latex]f[/latex]
indefinite integral
the most general antiderivative of [latex]f(x)[/latex] is the indefinite integral of [latex]f[/latex]; we use the notation [latex]\displaystyle\int f(x) dx[/latex] to denote the indefinite integral of [latex]f[/latex]
initial value problem
a problem that requires finding a function [latex]y[/latex] that satisfies the differential equation [latex]\frac{dy}{dx}=f(x)[/latex] together with the initial condition [latex]y(x_0)=y_0[/latex]