In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
s and
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
s. It is the counterpart to the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
for
differentiation, and can loosely be thought of as using the chain rule "backwards".
Substitution for a single variable
Introduction
Before stating the result
rigorously, consider a simple case using
indefinite integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
s.
Compute
.
Set
. This means
, or in
differential form,
. Now
:
where
is an arbitrary
constant of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
.
This procedure is frequently used, but not all integrals are of a form that permits its use. In any event, the result should be verified by differentiating and comparing to the original integrand.
:
For definite integrals, the limits of integration must also be adjusted, but the procedure is mostly the same.
Definite integrals
Let
be a
differentiable function with a
continuous derivative, where
is an
interval. Suppose that
is a
continuous function. Then
:
In Leibniz notation, the substitution
yields
:
Working heuristically with
infinitesimals yields the equation
:
which suggests the substitution formula above. (This equation may be put on a rigorous foundation by interpreting it as a statement about
differential forms.) One may view the method of integration by substitution as a partial justification of
Leibniz's notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just a ...
for integrals and derivatives.
The formula is used to transform one integral into another integral that is easier to compute. Thus, the formula can be read from left to right or from right to left in order to simplify a given integral. When used in the former manner, it is sometimes known as ''u''-substitution or ''w''-substitution in which a new variable is defined to be a function of the original variable found inside the
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
function multiplied by the derivative of the inner function. The latter manner is commonly used in
trigonometric substitution
In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities ...
, replacing the original variable with a
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
of a new variable and the original
differential with the differential of the trigonometric function.
Proof
Integration by substitution can be derived from the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
as follows. Let
and
be two functions satisfying the above hypothesis that
is continuous on
and
is integrable on the closed interval