Nonelementary integral

Definition

A nonelementary integral is an integral that cannot be expressed in terms of elementary functions such as polynomials, exponentials, logarithms, and trigonometric functions. These integrals often require special functions or numerical methods for their evaluation.

5 Must Know Facts For Your Next Test

1. Nonelementary integrals often appear in the context of Taylor series when dealing with complex functions.
2. Examples of nonelementary integrals include the integral of $e^{-x^2}$ and $\ln(x)$.
3. To solve nonelementary integrals, one might use power series expansions or special functions like the error function (erf).
4. Recognizing a nonelementary integral is crucial for applying appropriate techniques such as series expansion or numerical integration.
5. Some nonelementary integrals can be approximated using Taylor or Maclaurin series to find a polynomial approximation.

Review Questions

• What are nonelementary integrals and why can't they be expressed using elementary functions?
• How can you approach solving a nonelementary integral involving $e^{-x^2}$?
• Explain how Taylor series can be used to approximate nonelementary integrals.

Related terms

Elementary Functions: Functions composed of basic operations (addition, multiplication), exponential, logarithmic, trigonometric, and inverse trigonometric functions.

Taylor Series: An infinite sum of terms calculated from the values of a function’s derivatives at a single point. It approximates complex functions.

Error Function (erf): $\text{erf}(x)$ is a special function that arises in probability, statistics, and partial differential equations describing diffusion processes. It is used to evaluate some nonelementary integrals.