generalized integral

USAGE SUMMARY

The phrase "generalized integral" is correct and usable in written English.
It can be used in mathematical or scientific contexts to refer to an integral that has been extended or modified to apply to a broader class of functions or situations. Example: "In advanced calculus, we often encounter the concept of a generalized integral, which allows us to evaluate integrals that are not solvable using standard techniques."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

45 human-written examples

The associated direct problem is solved by the generalized integral transform technique.

Inverse Problems in Engineering Mechanics II International Symposium on Inverse Problems in Engineering Mechanics 2000 (ISIP 2000) Nagano, Japan

The resulting boundary value problem was solved analytically via the generalized integral transform technique (GITT).

Journal of Food Engineering

A generalized integral method is developed to analyze complex reactions in a catalyst pellet.

Chemical Engineering Science

The proposed approach sequentially calculates the concentration distributions of each species by employing only the generalized integral transform technique (GITT).

Advances in Water Resources

To solve the thermally developing problem, we use the generalized integral transform technique (GITT), also known as Sturm Liouville transform.

International Journal of Thermal Sciences

The generalized integral transform technique is performed with respect to the transformed system to obtain the analytical solution.

Journal of Hydrology

Human-verified similar examples from authoritative sources

Similar Expressions

15 human-written examples

A one-dimensional semi-analytical solution of land-derived solute transport, subject to tidal fluctuation in a coastal confined aquifer, was derived using the generalized integral-transform technique (GITT).

Journal of Hydrology

Many generalized fractional integral operators also take part in generalizing the theory of fractional integral operators [2, 3, 6, 8, 9, 13 15].

Journal of Inequalities and Applications

Motivated by the work in [13 15], firstly, we will prove a generalization of the identity given by Zhu et al. using generalized fractional integral operators.

Journal of Inequalities and Applications

Topics will include: ordinary and partial differential equations; generalized functions; integral transforms; Green's functions; nonlinear equations, chaos, and solitons; Hilbert space and linear operators; Feynman path integrals; Riemannian manifolds; tensor analysis; probability and statistics.

Columbia University

Note that in [2] a more general case of generalized fractional integrals was studied.

Journal of Inequalities and Applications

Expert writing Tips

Best practice

When using "generalized integral", ensure you clearly define which specific generalization you are referring to, as the term can encompass various extensions and modifications of the standard integral.

Common error

Avoid using "generalized integral" without specifying the type of generalization. Be explicit about whether you are referring to a Riemann-Stieltjes integral, Lebesgue integral, or another specific extension.

Linguistic Context

The phrase "generalized integral" functions as a noun phrase, where "generalized" acts as an adjective modifying the noun "integral". According to Ludwig AI, this term is correct and usable in written English. It refers to an extension or modification of the standard integral concept, as shown by the variety of examples provided.

Expression frequency: Common

Frequent in

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Ludwig's WRAP-UP

The phrase "generalized integral" is a grammatically sound and frequently employed term in scientific and mathematical discourse. According to Ludwig AI, it's considered accurate and suitable for use. It describes an integral that has been broadened or modified beyond its standard definition. The term's function is to denote a specific mathematical concept with a formal and scientific register. While numerous sources confirm its validity, ensuring specificity about the type of generalization is crucial for clarity. Alternative phrases include "extended integral" or "modified integral", though the original phrase is widely accepted within the scientific community.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

extended integral

Focuses on the expansion of the integral's applicability.

modified integral

Highlights alterations made to the standard integral.

integral generalization

Emphasizes the act of generalizing the integral.

generalized form of integral

Refers to a specific form that the integral takes in a generalized context.

integral extension

Similar to 'extended integral', but with a different word order.

integral in a generalized sense

Highlights that the integral is being used in a non-standard or broader way.

broader integral

Indicates a wider scope of integration.

abstract integral

Focuses on the abstract mathematical properties of the integral.

non-standard integral

Highlights that the integral deviates from the standard definition.

advanced integral

Suggests a more sophisticated or complex integral.

FAQs

How is a "generalized integral" different from a standard integral?

A "generalized integral" extends the concept of a standard integral to apply to a broader class of functions or situations. It might involve different integration techniques or definitions, such as the Lebesgue integral, which handles a wider range of functions than the Riemann integral.

When should I use the term "generalized integral"?

Use "generalized integral" when you are discussing an integral that goes beyond the basic Riemann integral taught in introductory calculus. This is common in advanced mathematical analysis and specific scientific applications.

What are some examples of "generalized integral" techniques?

Examples of "generalized integral" techniques include the Lebesgue integral, the Riemann-Stieltjes integral, and fractional integrals. These techniques broaden the types of functions that can be integrated and provide more powerful tools for mathematical analysis.

Are there alternatives to using the phrase "generalized integral"?

Depending on the context, you can use alternatives like "extended integral" or "modified integral". However, "generalized integral" is a standard term in mathematical literature.