#math/calculus An **antiderivative** of $f$ on an interval is a function $F$ such that $F'(x)=f(x)$. The family of all antiderivatives is the **indefinite integral** $ \int f(x)\,dx=F(x)+C, $ where $f$ is the **integrand** and $C$ is an arbitrary constant. Any two antiderivatives on the same interval differ by a constant. The basic rules reverse differentiation: $ \begin{aligned} \int[f(x)+g(x)]\,dx&=\int f(x)\,dx+\int g(x)\,dx,\\ \int af(x)\,dx&=a\int f(x)\,dx,\\ \int x^{n}\,dx&=\frac{x^{n+1}}{n+1}+C\qquad(n\neq -1). \end{aligned} $ Thus polynomials can be antidifferentiated term by term. **Example.** If $f(x)=2x^{2}+4$, then $ \int(2x^{2}+4)\,dx=\frac{2}{3}x^{3}+4x+C. $ Differentiating the result is the quickest check.