Integration by parts
A method for integrating a product by transferring differentiation from one factor to the other, derived from the product rule for derivatives.
Product rule: A differentiation rule stating that the derivative of a product is the derivative of one factor times the other, summed in both orders. Integration by parts reverses this rule and rearranges its terms.
Antiderivative: A function whose derivative equals a specified function on a given interval. The method requires finding an antiderivative for the factor designated as dv.
Integral of the natural logarithm: The antiderivative of the natural logarithm, obtained by integrating by parts. Writing ln x as ln x times 1 turns a function with no elementary derivative shortcut into an elementary antiderivative.
Trigonometric substitution: A change of variable using trigonometric functions to simplify radicals involving quadratic expressions. It simplifies radical structure by changing variables rather than redistributing differentiation.
Tabular integration: A structured technique that lists successive derivatives and integrals to organize repeated integration by parts. It streamlines repeated applications when one factor differentiates toward zero.
Indefinite integral: The family of antiderivatives of a function, conventionally written with an arbitrary constant. Its constant of integration must be included after applying the formula.
Integral of inverse tangent: The antiderivative of arctan x, found by treating it as a product with 1. Differentiating arctan x creates the rational function 1/(1+x²), making the remaining integral elementary.
Partial fraction decomposition: A method that rewrites a rational function as a sum of simpler rational fractions. It often integrates rational products directly, without choosing factors for integration by parts.
LIATE rule: A mnemonic that ranks logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions for choosing the differentiated factor. It offers a practical, though not infallible, guide for selecting the factor to differentiate.
Definite integral: A signed accumulation of a function over an interval, defined by a limit of sums or by the fundamental theorem of calculus. Its version of the method includes endpoint contributions from the boundary term.