Antiderivatives

(indefinite integrals)

Definition. Let the function $f:J\to \mathbb{R}$, with $J$ an interval. $F:J\to \mathbb{R}$ is an antiderivative of $f$ if $F$ is differentiable on $J$ and ${F}'(x)=f(x), (\forall )x\in J.$

Written: $\displaystyle \int{f(x)dx=}F(x)+C.$

Properties of antiderivatives:

1. $\displaystyle \int{[f(x)+g(x)]dx=}\int{f(x)dx+\int{g(x)}dx};$

2. $\displaystyle \int{a\cdot f(x)dx=a\cdot }\int{f(x)dx;}$

3. $\displaystyle \int{f(x)\cdot {g}'(x)dx=}f(x)g(x)-\int{{f}'(x)\cdot g(x)dx};$

A function continuous on an interval has antiderivatives on that interval.

The derivative of any differentiable function on an interval has the Darboux property (the intermediate value property).

If $f:I\to \mathbb{R}$ has antiderivatives on the interval $I$, then the function $f$ has the Darboux property on $I$.