Integrand¶

Definition¶

Given an integral of the form $\displaystyle \int_a^b f(x)\,\mathrm{d}x$ or $\displaystyle \int f(x)\,\mathrm{d}x$ , the function $f$ is called the integrand of the integral.

Interpretation¶

The integrand is the function being integrated. In $\int_a^b f(x)\,\mathrm{d}x$ it supplies the heights of the rectangles in each Riemann sum, and the integral adds these heights up, weighted by width. The letter $x$ in $f(x)$ is only a placeholder, and $\mathrm{d}x$ says which variable runs over the interval.

The integrand decides whether the integral exists. An integrand that is continuous or monotone on $[a,b]$ is Riemann integrable, while an unbounded one, such as $1/\sqrt{x}$ on $(0,1]$ , is not; integrals of that kind are treated later as improper integrals.

A common slip is to confuse the integrand with the result. In $\int_0^1 x^2\,\mathrm{d}x$ the integrand is $x^2$ , an antiderivative is $x^3/3$ , and the value of the integral is $\tfrac{1}{3}$ .

Key Results¶

Properties of the integrand influence integrability.

Examples¶

In $\displaystyle\int_0^1 x^2\,\mathrm{d}x\text{,}$ the integrand is $x^2\text{.}$
In $\displaystyle\int \,\mathrm{d}x\text{,}$ the integrand is $1\text{.}$

Prerequisites¶

Function

Sources¶

Math I: Calculus · Lecture 11

Symbols: integral, square root

Last updated: 20 October 2026 03:02