Approximating Areas
Antiderivatives and Indefinite Integrals - FTC1
Objectives
- Present a table of basic antiderivatives.
Summary
To find algebraic formulas for antiderivatives of more complicated algebraic functions, we need to think carefully about how we can reverse known differentiation rules. To that end, it is essential that we understand and recall known derivatives of basic functions, as well as the standard derivative rules.
The indefinite integral provides notation for antiderivatives. When we write "\(\int f(x) \, dx\)", we mean "the general antiderivative of \(f\)". In particular, if we have functions \(f\) and \(F\) such that \(F' = f \), the following two statements say the exact thing:
$$\frac{d}{dx} {[F(x)] = f(x)}\ \text{and} \int f(x) \,dx = F(x) + C $$ That is, \(f\) is the derivative of \(F\), and \(F\) is an antiderivative of \(f\).
- $$\int x^n \, dx = \frac{x^{n+1}}n +C$$
- $$\int \frac{1}{ax} \, dx = ln(ax) +C$$
- $$\int cosx \, dx = sinx +C$$
- $$\int sinx \, dx = -cosx +C$$
- $$\int sec^2{x} \, dx = tan{x} +C$$
- $$\int sec{x}tan{x} \, dx = sec{x} +C$$
- $$\int e^{ax} \, dx = \frac{e^{ax}}a +C$$