Indefinite Integral

This is the family of Antiderivative funcitons $F$ of a Continuous function $f$ s.t $\forall x\in I,F^{\prime }(x)=f(x)$

Definition

1. Suppose $f$ is Continuous on its domain
2. The indefinite integral is the family of Antiderivatives of $f$

Form

$\int f(x)dx=F(x)+C$ Where:

• $F(x)$ is the Antiderivative of $f(x)$
• $C$ is any constant

Properties

1. If $f(x),g(x)$ are Continuous functions on their domains, then $\int (f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$
2. $\forall c\in \mathbb{R},\int cf(x)dx=c\int f(x)dx$

Proofs

• Proving Indefinite Integral Property 2