Indefinite Integral

• (d1.12) $F$ is called an antiderivative of $f$ on an interval $I$, if $\forall x\in I,F^{\prime }(x)=f(x)$
  • The process of finding an antiderivative $F$ from its derivative $f=F^{\prime }$ is called integration (or antidifferentiation)
• (d2.1) The set of all antiderivatives called the indefinite integral of the $f$ with respect to $x$ and denote by:
  • $\int f(x)dx=\{F(x):F^{\prime }(x)=f(x),x\in I\}$
    • The symbol $\int$ is an integral sign
    • The function $f$ is the integrand of the integral
    • $x$ is the variable of integration
• (2.2) if $F$ is an antiderivative of $f$ then $\int f(x)dx=\{G:G=F+c,c\in \mathbb{R}\}$.
  • (or shortly, $F^{\prime }=f⟹\int f(x)dx=F(x)+C$)