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Integration by Parts definitions

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  • Integration by Parts
    A method for integrating products of functions when substitution fails, based on reversing the product rule for derivatives.
  • Product Rule
    A differentiation rule for finding the derivative of a product of two functions, foundational for deriving integration by parts.
  • u-Substitution
    A technique for simplifying integrals by substituting part of the integrand with a new variable, often ineffective for products.
  • Definite Integral
    An integral evaluated over specific bounds, requiring application of limits to all terms after integration by parts.
  • Indefinite Integral
    An integral without bounds, resulting in a general antiderivative plus a constant of integration.
  • Tabular Method
    An organized approach using a table of derivatives and integrals to streamline repeated integration by parts.
  • LIATE
    A mnemonic for choosing u in integration by parts: Logarithmic, Inverse, Algebraic, Trigonometric, Exponential.
  • Constant of Integration
    An arbitrary constant added to indefinite integrals, omitted in definite integrals.
  • Derivative
    A measure of how a function changes as its input changes, used in the product rule and integration by parts.
  • Antiderivative
    A function whose derivative is the given function, found through integration.
  • Bounds
    The upper and lower limits of integration, applied to all terms in definite integrals.
  • Integrand
    The function being integrated, often a product of two functions in integration by parts.
  • Repeated Integration by Parts
    A process where integration by parts is applied multiple times to fully evaluate an integral.
  • Power Rule
    A rule for differentiating or integrating powers of x, often used when finding derivatives in integration by parts.
  • Variable Substitution
    A method for simplifying integrals by changing variables, distinct from integration by parts.