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Substitution definitions

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  • Composite Function
    An expression where one function is nested inside another, often seen as f(g(x)), requiring special integration techniques.
  • Chain Rule
    A differentiation method for composite functions, whose reverse process inspires substitution in integration.
  • Substitution
    A technique for simplifying integrals by replacing a complex expression with a single variable, often called u.
  • U-Substitution
    A specific substitution method where a new variable u replaces an inner function, streamlining integration.
  • Change of Variables
    A process of rewriting an integral in terms of a new variable to simplify evaluation, especially for composite functions.
  • Definite Integral
    An integral with upper and lower bounds, representing the net area under a curve between two points.
  • Indefinite Integral
    An integral without bounds, representing a family of antiderivatives plus a constant of integration.
  • Bounds
    The limits of integration in a definite integral, which may be transformed during substitution.
  • Antiderivative
    A function whose derivative yields the original integrand, found during the integration process.
  • Power Rule
    A rule for integrating or differentiating expressions with exponents, crucial for evaluating polynomials.
  • Constant of Integration
    An arbitrary constant added to indefinite integrals, accounting for all possible antiderivatives.
  • Reciprocal
    A value that, when multiplied by a given number, yields one; used to adjust constants during substitution.
  • Fundamental Theorem of Calculus
    A principle linking differentiation and integration, allowing evaluation of definite integrals using antiderivatives.
  • Placeholder Variable
    A temporary symbol, such as u, used to simplify expressions during substitution before reverting to the original variable.
  • Derivative
    A measure of how a function changes as its input changes, used to find du when making a substitution.