Integral vs. Integrand — What's the Difference?

In calculus, a tool for finding quantities like areas, volumes, and accumulations.

It specifies the variable of integration when included in an integral.

Integral calculus is used alongside differential calculus in many scientific fields.

The function f(x) placed within the integral for integration.

A mathematical operation that sums up the values of a function over an interval.

Can consist of simple or complex expressions.

Integrals are solved using various techniques depending on the integrand.

It remains unchanged whether the integral is definite or indefinite.

It can be "definite" giving a specific result or "indefinite" providing a function.

Understanding its behavior is key to choosing the right integration technique.

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration.

A function to be integrated.

Necessary to make a whole complete; essential or fundamental

(calculus) The function that is to be integrated

Of or denoted by an integer.

A function of which a given function is the derivative, i.e. which yields that function when differentiated, and which may express the area under the curve of a graph of the function.

Essential or necessary for completeness; constituent

Possessing everything essential; entire.

Expressed or expressible as or in terms of integers.

Expressed as or involving integrals.

A complete unit; a whole.

A number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or areal measurement of that unit, and all such products are summed.

A definite integral.

An indefinite integral.

Constituting a whole together with other parts or factors; not omittable or removable

(mathematics) Of, pertaining to, or being an integer.

(mathematics) Relating to integration.

(obsolete) Whole; undamaged.

(mathematics) One of the two fundamental operations of calculus (the other being differentiation), whereby a function's displacement, area, volume, or other qualities arising from the study of infinitesimal change are quantified, usually defined as a limiting process on a sequence of partial sums. Denoted using a long s: ∫, or a variant thereof.

(specifically) Any of several analytic formalizations of this operation: the Riemann integral, the Lebesgue integral, etc.

(mathematics) A definite integral: the result of the application of such an operation onto a function and a suitable subset of the function's domain: either a number or positive or negative infinity. In the former case, the integral is said to be finite or to converge; in the latter, the integral is said to diverge. In notation, the domain of integration is indicated either below the sign, or, if it is an interval, with its endpoints as sub- and super-scripts, and the function being integrated forming part of the integrand (or, generally, differential form) appearing in front of the integral sign.

(mathematics) An indefinite integral: the result of the application of such an operation onto a function together with an indefinite domain, yielding a function; a function's antiderivative;

The fluent of a given fluxion in Newtonian calculus.

Lacking nothing of completeness; complete; perfect; uninjured; whole; entire.

Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.

Of, pertaining to, or being, a whole number or undivided quantity; not fractional.

A whole; an entire thing; a whole number; an individual.

An expression which, being differentiated, will produce a given differential. See differential Differential, and Integration. Cf. Fluent.

The result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)

Existing as an essential constituent or characteristic;

Constituting the undiminished entirety; lacking nothing essential especially not damaged;

Correctly identifying the integrand is crucial because it defines the function to be integrated and influences the selection of integration methods and the interpretation of the result.

No, integrands can be any function that is mathematically integrable, including algebraic, trigonometric, exponential, logarithmic functions, and more.

The choice of differential in an integral, such as dx, dy, or dt, depends on the variable of integration which is specified by the integrand.

Yes, integrals can involve expressions with multiple functions multiplied together or functions composed within other functions, which are then integrated as a single integrand.

Indefinite integrals and antiderivatives are essentially the same; both involve finding a function whose derivative is the integrand, without specifying limits of integration.

In multivariable integrals, integrands involve more than one variable and are integrated with respect to each variable sequentially or simultaneously, depending on the integral setup.

Techniques such as substitution, integration by parts, partial fractions, and numerical integration are used to handle complex integrands.

Integration by parts is a technique that uses the product rule for differentiation to integrate products of functions, breaking down complex integrands into simpler parts.

The fundamental theorem of calculus links the concept of differentiation and integration, stating that if a function is continuous over an interval, the integral of its derivative over that interval is equal to the difference in the values of the original function at the endpoints of the interval.

No, the integral of a constant is not zero; it is the constant multiplied by the variable of integration plus a constant of integration, reflecting a linear relationship.

While many functions can serve as integrands, they must be integrable over the chosen interval; some functions might not be integrable due to discontinuities or other issues.

Numerical methods can be applied to most types of integrands, especially when analytical methods are too complex or impossible, but they require careful consideration of error margins and computational resources.

The limits in a definite integral specify the range over which the integrand is to be integrated, determining the starting and ending points of the accumulation process.

When an integrand changes sign, it can lead to the integral representing areas above and below the axis, effectively subtracting these areas, which is important in applications like computing net changes.

Symmetry in an integrand can simplify the process of integration, as symmetric functions often lead to cancellations or reductions that make the integral easier to evaluate.