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 ...
, and more generally in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, integration by parts or partial integration is a process that finds the
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 ...
of a
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of
functions in terms of the integral of the product of their
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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 ...
. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
of
differentiation.
The integration by parts formula states:
Or, letting
and
while
and
, the formula can be written more compactly:
Mathematician
Brook Taylor discovered integration by parts, first publishing the idea in 1715.
More general formulations of integration by parts exist for the
Riemann–Stieltjes and
Lebesgue–Stieltjes integrals. The
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
analogue for
sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
is called
summation by parts
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformat ...
.
Theorem
Product of two functions
The theorem can be derived as follows. For two
continuously differentiable functions ''u''(''x'') and ''v''(''x''), the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
states:
Integrating both sides with respect to ''x'',
and noting that an
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 ...
is an antiderivative gives
where we neglect writing the
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 yields the formula for integration by parts:
or in terms of the
differentials ,
This is to be understood as an equality of functions with an unspecified constant added to each side. Taking the difference of each side between two values ''x'' = ''a'' and ''x'' = ''b'' and applying 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 ...
gives the definite integral version:
The original integral ∫ ''uv''′ ''dx'' contains the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
''v''′; to apply the theorem, one must find ''v'', the
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 ...
of ''v''
', then evaluate the resulting integral ∫ ''vu''′ ''dx''.
Validity for less smooth functions
It is not necessary for ''u'' and ''v'' to be continuously differentiable. Integration by parts works if ''u'' is
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
and the function designated ''v''′ is
Lebesgue integrable
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
(but not necessarily continuous). (If ''v''′ has a point of discontinuity then its antiderivative ''v'' may not have a derivative at that point.)
If the interval of integration is not
compact, then it is not necessary for ''u'' to be absolutely continuous in the whole interval or for ''v''′ to be Lebesgue integrable in the interval, as a couple of examples (in which ''u'' and ''v'' are continuous and continuously differentiable) will show. For instance, if
''u'' is not absolutely continuous on the interval , but nevertheless
so long as
is taken to mean the limit of
as
and so long as the two terms on the right-hand side are finite. This is only true if we choose
Similarly, if
''v''′ is not Lebesgue integrable on the interval , but nevertheless
with the same interpretation.
One can also easily come up with similar examples in which ''u'' and ''v'' are ''not'' continuously differentiable.
Further, if
is a function of bounded variation on the segment
and
is differentiable on
then
where
denotes the signed measure corresponding to the function of bounded variation
, and functions
are extensions of
to
which are respectively of bounded variation and differentiable.
Product of many functions
Integrating the product rule for three multiplied functions, ''u''(''x''), ''v''(''x''), ''w''(''x''), gives a similar result:
In general, for ''n'' factors
which leads to
Visualization
Consider a parametric curve by (''x'', ''y'') = (''f''(''t''), ''g''(''t'')). Assuming that the curve is locally
one-to-one and
integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
, we can define
:
:
The area of the blue region is
:
Similarly, the area of the red region is
:
The total area ''A''
1 + ''A''
2 is equal to the area of the bigger rectangle, ''x''
2''y''
2, minus the area of the smaller one, ''x''
1''y''
1:
:
Or, in terms of ''t'',
:
Or, in terms of indefinite integrals, this can be written as
:
Rearranging:
:
Thus integration by parts may be thought of as deriving the area of the blue region from the area of rectangles and that of the red region.
This visualization also explains why integration by parts may help find the integral of an inverse function ''f''
−1(''x'') when the integral of the function ''f''(''x'') is known. Indeed, the functions ''x''(''y'') and ''y''(''x'') are inverses, and the integral ∫ ''x'' ''dy'' may be calculated as above from knowing the integral ∫ ''y'' ''dx''. In particular, this explains use of integration by parts to integrate
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
and
inverse trigonometric functions. In fact, if
is a differentiable one-to-one function on an interval, then integration by parts can be used to derive a formula for the integral of
in terms of the integral of
. This is demonstrated in the article,
Integral of inverse functions
In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f^ of a continuous and invertible function in terms of f^ and an antiderivative of This formula was publishe ...
.
Applications
Finding antiderivatives
Integration by parts is a
heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions ''u''(''x'')''v''(''x'') such that the residual integral from the integration by parts formula is easier to evaluate than the single function. The following form is useful in illustrating the best strategy to take:
:
On the right-hand side, ''u'' is differentiated and ''v'' is integrated; consequently it is useful to choose ''u'' as a function that simplifies when differentiated, or to choose ''v'' as a function that simplifies when integrated. As a simple example, consider:
:
Since the derivative of ln(''x'') is , one makes (ln(''x'')) part ''u''; since the antiderivative of is −, one makes ''dx'' part ''dv''. The formula now yields:
:
The antiderivative of − can be found with the
power rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated usin ...
and is .
Alternatively, one may choose ''u'' and ''v'' such that the product ''u''′ (∫''v'' ''dx'') simplifies due to cancellation. For example, suppose one wishes to integrate:
:
If we choose ''u''(''x'') = ln(, sin(''x''), ) and ''v''(''x'') = sec
2x, then ''u'' differentiates to 1/ tan ''x'' using 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 , ...
and ''v'' integrates to tan ''x''; so the formula gives:
:
The integrand simplifies to 1, so the antiderivative is ''x''. Finding a simplifying combination frequently involves experimentation.
In some applications, it may not be necessary to ensure that the integral produced by integration by parts has a simple form; for example, in
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, it may suffice that it has small magnitude and so contributes only a small error term. Some other special techniques are demonstrated in the examples below.
Polynomials and trigonometric functions
In order to calculate
:
let:
:
:
then:
:
where ''C'' is a
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 ...
.
For higher powers of ''x'' in the form
:
repeatedly using integration by parts can evaluate integrals such as these; each application of the theorem lowers the power of ''x'' by one.
Exponentials and trigonometric functions
An example commonly used to examine the workings of integration by parts is
:
Here, integration by parts is performed twice. First let
:
:
then:
:
Now, to evaluate the remaining integral, we use integration by parts again, with:
:
:
Then:
:
Putting these together,
:
The same integral shows up on both sides of this equation. The integral can simply be added to both sides to get
:
which rearranges to
:
where again ''C'' (and ''C''′ = ''C''/2) is a
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 ...
.
A similar method is used to find the
integral of secant cubed
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
:\begin
\int \sec^3 x \, dx
&= \tfrac12\sec x \tan x + \tfrac12 \int \sec x\, dx + C \\ mu&= \tfrac12(\sec x \tan x + \ln \left, \sec x + \ta ...
.
Functions multiplied by unity
Two other well-known examples are when integration by parts is applied to a function expressed as a product of 1 and itself. This works if the derivative of the function is known, and the integral of this derivative times ''x'' is also known.
The first example is ∫ ln(''x'') d''x''. We write this as:
:
Let:
:
:
then:
:
where ''C'' is the
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 ...
.
The second example is the
inverse tangent function arctan(''x''):
:
Rewrite this as
:
Now let:
:
:
then
:
using a combination of the
inverse chain rule method
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
and the
natural logarithm integral condition.
LIATE rule
A rule of thumb has been proposed, consisting of choosing as ''u'' the function that comes first in the following list:
:L –
logarithmic functions:
etc.
:I –
inverse trigonometric functions (including
hyperbolic analogues):
etc.
:A –
algebraic functions:
etc.
:T –
trigonometric functions (including
hyperbolic analogues):
etc.
:E –
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
s:
etc.
The function which is to be ''dv'' is whichever comes last in the list. The reason is that functions lower on the list generally have easier
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 than the functions above them. The rule is sometimes written as "DETAIL" where ''D'' stands for ''dv'' and the top of the list is the function chosen to be ''dv''.
To demonstrate the LIATE rule, consider the integral
:
Following the LIATE rule, ''u'' = ''x'', and ''dv'' = cos(''x'') ''dx'', hence ''du'' = ''dx'', and ''v'' = sin(''x''), which makes the integral become
:
which equals
:
In general, one tries to choose ''u'' and ''dv'' such that ''du'' is simpler than ''u'' and ''dv'' is easy to integrate. If instead cos(''x'') was chosen as ''u'', and ''x dx'' as ''dv'', we would have the integral
:
which, after recursive application of the integration by parts formula, would clearly result in an infinite recursion and lead nowhere.
Although a useful rule of thumb, there are exceptions to the LIATE rule. A common alternative is to consider the rules in the "ILATE" order instead. Also, in some cases, polynomial terms need to be split in non-trivial ways. For example, to integrate
:
one would set
:
so that
:
Then
:
Finally, this results in
:
Integration by parts is often used as a tool to prove theorems in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
.
Wallis product
The Wallis infinite product for
:
may be
derived using integration by parts.
Gamma function identity
The
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
is an example of a
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined b ...
, defined as an
improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
for
. Integration by parts illustrates it to be an extension of the factorial function:
:
Since
:
when
is a natural number, that is,
, applying this formula repeatedly gives the
factorial:
Use in harmonic analysis
Integration by parts is often used in
harmonic analysis, particularly
Fourier analysis, to show
that quickly oscillating integrals with sufficiently smooth integrands decay quickly. The most common example of this is its use in showing that the decay of function's Fourier transform depends on the smoothness of that function, as described below.
Fourier transform of derivative
If ''f'' is a ''k''-times continuously differentiable function and all derivatives up to the ''k''th one decay to zero at infinity, then its
Fourier transform satisfies
:
where is the ''k''th derivative of ''f''. (The exact constant on the right depends on the
convention of the Fourier transform used.) This is proved by noting that
:
so using integration by parts on the Fourier transform of the derivative we get
:
Applying this
inductively gives the result for general ''k''. A similar method can be used to find the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
of a derivative of a function.
Decay of Fourier transform
The above result tells us about the decay of the Fourier transform, since it follows that if ''f'' and are integrable then
:
In other words, if ''f'' satisfies these conditions then its Fourier transform decays at infinity at least as quickly as . In particular, if then the Fourier transform is integrable.
The proof uses the fact, which is immediate from the
definition of the Fourier transform, that
:
Using the same idea on the equality stated at the start of this subsection gives
:
Summing these two inequalities and then dividing by gives the stated inequality.
Use in operator theory
One use of integration by parts in
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
is that it shows that the (where ∆ is the
Laplace operator) is a
positive operator In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \l ...
on (see
''L''''p'' space). If ''f'' is smooth and compactly supported then, using integration by parts, we have
:
Other applications
* Determining
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s in
Sturm–Liouville theory
* Deriving the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
in the
calculus of variations
Repeated integration by parts
Considering a second derivative of
in the integral on the LHS of the formula for partial integration suggests a repeated application to the integral on the RHS:
:
Extending this concept of repeated partial integration to derivatives of degree leads to
:
This concept may be useful when the successive integrals of
are readily available (e.g., plain exponentials or sine and cosine, as in
Laplace or
Fourier transforms), and when the th derivative of
vanishes (e.g., as a polynomial function with degree
). The latter condition stops the repeating of partial integration, because the RHS-integral vanishes.
In the course of the above repetition of partial integrations the integrals
:
and
and
get related. This may be interpreted as arbitrarily "shifting" derivatives between
and
within the integrand, and proves useful, too (see
Rodrigues' formula
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out ...
).
Tabular integration by parts
The essential process of the above formula can be summarized in a table; the resulting method is called "tabular integration" and was featured in the film ''
Stand and Deliver'' (1988).
For example, consider the integral
:
and take
Begin to list in column A the function
and its subsequent derivatives
until zero is reached. Then list in column B the function
and its subsequent integrals
until the size of column B is the same as that of column A. The result is as follows:
:
The product of the entries in of columns A and B together with the respective sign give the relevant integrals in in the course of repeated integration by parts. yields the original integral. For the complete result in the must be added to all the previous products () of the of column A and the of column B (i.e., multiply the 1st entry of column A with the 2nd entry of column B, the 2nd entry of column A with the 3rd entry of column B, etc. ...) with the given This process comes to a natural halt, when the product, which yields the integral, is zero ( in the example). The complete result is the following (with the alternating signs in each term):
:
This yields
:
The repeated partial integration also turns out useful, when in the course of respectively differentiating and integrating the functions
and
their product results in a multiple of the original integrand. In this case the repetition may also be terminated with this index This can happen, expectably, with exponentials and trigonometric functions. As an example consider
:
:
In this case the product of the terms in columns A and B with the appropriate sign for index yields the negative of the original integrand (compare
:
Observing that the integral on the RHS can have its own constant of integration
, and bringing the abstract integral to the other side, gives
:
and finally:
:
where ''C'' = ''C''′/2.
Higher dimensions
Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function ''u'' and vector-valued function (vector field) V.
The
product rule for divergence states:
Suppose
is an
open bounded subset of
with a
piecewise smooth
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
. Integrating over
with respect to the standard volume form
, and applying the
divergence theorem, gives:
where
is the outward unit normal vector to the boundary, integrated with respect to its standard Riemannian volume form
. Rearranging gives:
or in other words
The
regularity requirements of the theorem can be relaxed. For instance, the boundary
need only be
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
, and the functions ''u'', ''v'' need only lie in the
Sobolev space ''H''
1(Ω).
Green's first identity
Consider the continuously differentiable vector fields
and
, where
is the ''i''-th standard basis vector for
. Now apply the above integration by parts to each
times the vector field
:
Summing over ''i'' gives a new integration by parts formula:
The case
, where
, is known as the first of
Green's identities:
See also
*
Integration by parts for the Lebesgue–Stieltjes integral
*
Integration by parts for
semimartingale
In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...
s, involving their quadratic covariation.
*
Integration by substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
*
Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
Notes
Further reading
*
*
*
*
External links
*
Integration by parts—from MathWorld
{{Integrals
Integral calculus
Mathematical identities
Theorems in analysis
Theorems in calculus
es:Métodos de integración#Método de integración por partes