# Sum rule

Theorem: If $F(x)$ is an antiderivative of $f(x)$, and $G(x)$ is an antiderivative of $g(x)$, then $F(x)+G(x)$ is an antiderivative of $f(x)+g(x)$.

Remark: The sum rule can be extended to the sum of three or more functions.

Example
Let $f(x)=3x^4$, $g(x)=e^{3x}$ and $h(x)=\frac{1}{x}$.
From the table with antiderivatives of elementary functions it follows that $F(x)=\frac{3}{5}x^5$, $G(x)=\frac{1}{3}e^{3x}$ and $H(x)=\ln{x}$ are antiderivatives of $f(x)$, $g(x)$ and $h(x)$, respectively.
The sum rule of antidifferentiation dictates for instance that
• $F(x)+G(x)+H(x)$ is an antiderivative of $f(x)+g(x)+h(x)$;
• $F(x)+H(x)$ is an antiderivative of $f(x)+h(x)$;
• $G(x)+H(x)$ is an antiderivative of $g(x)+h(x)$.