Question about nonelementary integrals

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In summary, implicit functions and integrals are two separate concepts. When taking the derivative of an implicit function, the resulting expression will include both the independent and dependent variables. However, you may be able to manipulate it to get an explicit expression. Integrating an explicit function may or may not result in an expression that can be expressed in terms of elementary functions, and this is unrelated to implicit functions.
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DivergentSpectrum
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are nonelementary integrals implicit functions?

ie, when we do implicit differentation, we get an explicit function. What if i go the opposite way, and integrate an explicit function to get an implicit antiderivative?
 
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You seem to be mixing up two unrelated concepts.

When you get the derivative of an implicit function, at first the expression for the derivative will include both the independent variable and the dependent variable. You may be able to manipulate it to get an explicit expression for the derivative.

When you integrate an explicit function the integral may or may be expressible in terms of elementary functions. Implicit has nothing to do with it.
 

FAQ: Question about nonelementary integrals

What is a nonelementary integral?

A nonelementary integral is an integral that cannot be expressed in terms of elementary functions (such as polynomials, exponential, trigonometric, and logarithmic functions). These integrals often involve special functions or transcendental functions.

How do you solve a nonelementary integral?

Solving a nonelementary integral requires using advanced techniques, such as integration by parts, substitution, or trigonometric identities. In some cases, it may also require the use of specialized methods, such as contour integration or series expansions.

What are some examples of nonelementary integrals?

Some examples of nonelementary integrals include the Gaussian integral, the error function, the Fresnel integral, and the logarithmic integral. These integrals often arise in physics, engineering, and statistics.

Why are nonelementary integrals important?

Nonelementary integrals play a crucial role in many areas of mathematics and science. They are used to solve differential equations, evaluate complex functions, and analyze real-world problems. In addition, many challenging mathematical concepts and techniques can be explored through the study of nonelementary integrals.

Are there any strategies for identifying nonelementary integrals?

Some strategies for identifying nonelementary integrals include looking for special functions in the integrand, checking for transcendental or non-algebraic functions, and considering the complexity of the integrand. In some cases, it may also be helpful to consult a table of known integrals or use computer software to determine the type of integral.

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