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Integrand

Integrand

The function being integrated in either a definite or indefinite integral.

 

Example: x2cos 3x is the integrand in ∫ x2cos 3x dx.

Key Formula

∫abf(x) dx\int_a^b f(x)\,dx
Where:
  • f(x)f(x) = The integrand — the function being integrated
  • aa = Lower limit of integration (for a definite integral)
  • bb = Upper limit of integration (for a definite integral)
  • dxdx = The differential, indicating integration with respect to x

Worked Example

Problem: Identify the integrand in the integral ∫035x2 dx\int_0^3 5x^2\,dx and then evaluate the integral.
Step 1: Identify the integrand. The integrand is the function between the integral sign and the differential dxdx.
Integrand=5x2\text{Integrand} = 5x^2
Step 2: Find the antiderivative of the integrand 5x25x^2.
∫5x2 dx=5x33+C\int 5x^2\,dx = \frac{5x^3}{3} + C
Step 3: Evaluate the antiderivative at the limits of integration using the Fundamental Theorem of Calculus.
[5x33]03=5(3)33−5(0)33=1353−0=45\left[\frac{5x^3}{3}\right]_0^3 = \frac{5(3)^3}{3} - \frac{5(0)^3}{3} = \frac{135}{3} - 0 = 45
Answer: The integrand is 5x25x^2, and the value of the definite integral is 4545.

Another Example

Problem: Identify the integrand in ∫(ex+3sin⁡x) dx\int (e^x + 3\sin x)\,dx.
Step 1: Look at everything between the integral sign ∫\int and the differential dxdx. That entire expression is the integrand.
Integrand=ex+3sin⁡x\text{Integrand} = e^x + 3\sin x
Step 2: Note that the integrand here is a sum of two terms. The whole expression ex+3sin⁡xe^x + 3\sin x is the integrand, not just one piece of it.
Answer: The integrand is ex+3sin⁡xe^x + 3\sin x. It includes every term inside the integral before dxdx.

Frequently Asked Questions

What is the difference between the integrand and the integral?
The integrand is the function being integrated — it is the input to the integration process. The integral is the entire expression including the integral sign, limits, integrand, and differential, or it can refer to the result (the antiderivative or numerical value) of performing the integration. Think of the integrand as an ingredient and the integral as the finished product.
Is dxdx part of the integrand?
No. The differential dxdx indicates the variable of integration but is not considered part of the integrand. The integrand is only the function that sits between the integral sign (and any limits) and the differential.

Integrand vs. Integral

The integrand is the function you feed into an integral. The integral is the operation itself (or its result). For example, in ∫01x3 dx\int_0^1 x^3\,dx, the integrand is x3x^3, while the integral equals 14\frac{1}{4}. The integrand names a part; the integral names the whole process or outcome.

Why It Matters

Correctly identifying the integrand is the first step in choosing an integration technique — substitution, integration by parts, or partial fractions all depend on the structure of the integrand. When integrands become complex, rewriting or simplifying them before integrating can save significant effort. The term also appears constantly in physics and engineering, where setting up the correct integrand determines whether a model for area, volume, work, or probability is valid.

Common Mistakes

Mistake: Including dxdx as part of the integrand.
Correction: The differential dxdx tells you which variable you integrate with respect to; it is separate from the integrand. In ∫3x2 dx\int 3x^2\,dx, the integrand is 3x23x^2 alone.
Mistake: Identifying only part of a sum or product as the integrand.
Correction: The integrand is the entire expression between ∫\int (with limits, if any) and dxdx. In ∫(x2+7x−1) dx\int (x^2 + 7x - 1)\,dx, the full integrand is x2+7x−1x^2 + 7x - 1, not just one of those terms.

Related Terms