1. Find the antiderivative of the function $f(x)=4 x^{3}$.
Solution. Using the differentiation rule, one can guess that on the interval $(-\infty, \infty)$, the antiderivative is $F(x)=x^{4}$. Indeed, $F^{\prime}(x)=4 x^{3}$ for all $x \in(-\infty, \infty)$.

Remark. If it is said that $F(x)$ is an antiderivative of the function $f(x)$, but the interval is not specified, then in the further context, this interval will be assumed to be any interval where the function $f(x)$ is defined.