integrand

The integrand appears in the integral sign.

Definition

1. Noun:
   • In mathematics: "integrand" is a noun referring to a function that is to be integrated. Specifically, it is the expression or function placed under the integral sign (∫) in calculus.

Usage Examples

• (The function being integrated is x squared.)
• (Make the expression under the integral sign easier to work with.)
• (The function being integrated must be continuous.)

Advanced Usage

• "Improper integrand": an integrand that becomes infinite within the interval of integration.

  • The integrand 1/(x-1) has a singularity at x=1, making it an improper integrand. (The function is not well-defined at that point.)

• "Analytic integrand": an integrand that is analytic (smooth and complex-differentiable) over the domain.

  • For contour integration, the integrand must be analytic except at isolated poles. (The function must be smooth in the complex plane.)

Variants and Related Words

• Integrand (n): the function being integrated. No common variants, but related terms include:
  • Integral (n): the result of the integration process.

    • The integral of x² is (1/3)x³ + C. (The result of integrating the integrand.)
  • Integration (n): the process of finding an integral.

    • Integration of the integrand requires substitution. (The process of integrating the function.)

Synonyms

• Function to be integrated: a descriptive phrase, not a direct synonym.
• Integrable function: a function that can be integrated (often used synonymously in context).

Related Idioms

• "Integrand is well-behaved": an informal expression meaning the integrand is continuous and finite over the integration interval.

  • If the integrand is well-behaved, the integral is easy to compute. (The function has no singularities or discontinuities.)

• "Integrand blows up": an informal phrase meaning the integrand approaches infinity at some point.

  • The integrand blows up at x=0, so we must use improper integration. (The function becomes infinite.)