Mathematical method in calculus
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. 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 of differentiation; it is indeed derived using the product rule.
The integration by parts formula states:
formula_1
Or, letting formula_2 and formula_3 while formula_4 and formula_5 the formula can be written more compactly:
formula_6
The former expression is written as a definite integral and the latter is written as an indefinite integral. Applying the appropriate limits to the latter expression should yield the former, but the latter is not necessarily equivalent to the former.
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 analogue for sequences is called summation by parts.
Theorem.
Product of two functions.
The theorem can be derived as follows. For two continuously differentiable functions formula_7 and formula_8, the product rule states:
formula_9
Integrating both sides with respect to formula_10,
formula_11
and noting that an indefinite integral is an antiderivative gives
formula_12
where we neglect writing the constant of integration. This yields the formula for integration by parts:
formula_13
or in terms of the differentials formula_14, formula_15
formula_16
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 formula_17 and formula_18 and applying the fundamental theorem of calculus gives the definite integral version:
formula_19
The original integral formula_20 contains the derivative v'; to apply the theorem, one must find v, the antiderivative of v', then evaluate the resulting integral formula_21
Validity for less smooth functions.
It is not necessary for formula_22 and formula_23 to be continuously differentiable. Integration by parts works if formula_22 is absolutely continuous and the function designated formula_25 is Lebesgue integrable (but not necessarily continuous). (If formula_25 has a point of discontinuity then its antiderivative formula_23 may not have a derivative at that point.)
If the interval of integration is not compact, then it is not necessary for formula_22 to be absolutely continuous in the whole interval or for formula_25 to be Lebesgue integrable in the interval, as a couple of examples (in which formula_22 and formula_23 are continuous and continuously differentiable) will show. For instance, if
formula_32
formula_22 is not absolutely continuous on the interval [1, ∞), but nevertheless:
formula_34
so long as formula_35 is taken to mean the limit of formula_36 as formula_37 and so long as the two terms on the right-hand side are finite. This is only true if we choose formula_38 Similarly, if
formula_39
formula_25 is not Lebesgue integrable on the interval [1, ∞), but nevertheless
formula_34
with the same interpretation.
One can also easily come up with similar examples in which formula_22 and formula_23 are "not" continuously differentiable.
Further, if formula_44 is a function of bounded variation on the segment formula_45 and formula_46 is differentiable on formula_45 then
formula_48
where formula_49 denotes the signed measure corresponding to the function of bounded variation formula_50, and functions formula_51 are extensions of formula_52 to formula_53 which are respectively of bounded variation and differentiable.
Product of many functions.
Integrating the product rule for three multiplied functions, formula_7, formula_8, formula_56, gives a similar result:
formula_57
In general, for formula_58 factors
formula_59
which leads to
formula_60
Visualization.
Consider a parametric curve formula_61. Assuming that the curve is locally one-to-one and integrable, we can define
formula_62
The area of the blue region is
formula_63
Similarly, the area of the red region is
formula_64
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:
formula_65
Or, in terms of "t",
formula_66
Or, in terms of indefinite integrals, this can be written as
formula_67
Rearranging:
formula_68
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 and inverse trigonometric functions. In fact, if formula_69 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 formula_70 in terms of the integral of formula_69. This is demonstrated in the article, Integral of inverse functions.
Applications.
Finding antiderivatives.
Integration by parts is a heuristic 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:
formula_72
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:
formula_73
Since the derivative of ln("x") is , one makes (ln("x")) part "u"; since the antiderivative of is −, one makes part "v". The formula now yields:
formula_74
The antiderivative of − can be found with the power rule 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:
formula_75
If we choose "u"("x") = ln(|sin("x")|) and "v"("x") = sec2x, then "u" differentiates to formula_76 using the chain rule and "v" integrates to tan "x"; so the formula gives:
formula_77
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, 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
formula_78
let:
formula_79
then:
formula_80
where "C" is a constant of integration.
For higher powers of formula_10 in the form
formula_82
repeatedly using integration by parts can evaluate integrals such as these; each application of the theorem lowers the power of formula_10 by one.
Exponentials and trigonometric functions.
An example commonly used to examine the workings of integration by parts is
formula_84
Here, integration by parts is performed twice. First let
formula_85
then:
formula_86
Now, to evaluate the remaining integral, we use integration by parts again, with:
formula_87
Then:
formula_88
Putting these together,
formula_89
The same integral shows up on both sides of this equation. The integral can simply be added to both sides to get
formula_90
which rearranges to
formula_91
where again formula_92 (and formula_93) is a constant of integration.
A similar method is used to find the integral of secant cubed.
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 formula_10 is also known.
The first example is formula_95. We write this as:
formula_96
Let:
formula_97
formula_98
then:
formula_99
where formula_92 is the constant of integration.
The second example is the inverse tangent function formula_101:
formula_102
Rewrite this as
formula_103
Now let:
formula_104
formula_98
then
formula_106
using a combination of the inverse chain rule method and the natural logarithm integral condition.
LIATE rule.
The LIATE rule is a rule of thumb for integration by parts. It involves choosing as "u" the function that comes first in the following list:
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 simpler antiderivatives 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". An alternative to this rule is the ILATE rule, where inverse trigonometric functions come before logarithmic functions.
To demonstrate the LIATE rule, consider the integral
formula_112
Following the LIATE rule, "u" = "x", and "dv" = cos("x") "dx", hence "du" = "dx", and "v" = sin("x"), which makes the integral become
formula_113
which equals
formula_114
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
formula_115
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
formula_116
one would set
formula_117
so that
formula_118
Then
formula_119
Finally, this results in
formula_120
Integration by parts is often used as a tool to prove theorems in mathematical analysis.
Wallis product.
The Wallis infinite product for formula_121
formula_122
may be derived using integration by parts.
Gamma function identity.
The gamma function is an example of a special function, defined as an improper integral for formula_123. Integration by parts illustrates it to be an extension of the factorial function:
formula_124
Since
formula_125
when formula_126 is a natural number, that is, formula_127, applying this formula repeatedly gives the factorial: formula_128
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 formula_69 is a formula_130-times continuously differentiable function and all derivatives up to the formula_130th one decay to zero at infinity, then its Fourier transform satisfies
formula_132
where formula_133 is the formula_130th derivative of formula_69. (The exact constant on the right depends on the convention of the Fourier transform used.) This is proved by noting that
formula_136
so using integration by parts on the Fourier transform of the derivative we get
formula_137
Applying this inductively gives the result for general formula_130. A similar method can be used to find the Laplace transform 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 formula_69 and formula_133 are integrable then
formula_141
In other words, if formula_69 satisfies these conditions then its Fourier transform decays at infinity at least as quickly as 1/|"ξ"|"k". In particular, if formula_143 then the Fourier transform is integrable.
The proof uses the fact, which is immediate from the definition of the Fourier transform, that
formula_144
Using the same idea on the equality stated at the start of this subsection gives
formula_145
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 is that it shows that the −∆ (where ∆ is the Laplace operator) is a positive operator on formula_146 (see "L""p" space). If formula_69 is smooth and compactly supported then, using integration by parts, we have
formula_148
Repeated integration by parts.
Considering a second derivative of formula_23 in the integral on the LHS of the formula for partial integration suggests a repeated application to the integral on the RHS:
formula_150
Extending this concept of repeated partial integration to derivatives of degree n leads to
formula_151
This concept may be useful when the successive integrals of formula_152 are readily available (e.g., plain exponentials or sine and cosine, as in Laplace or Fourier transforms), and when the nth derivative of formula_22 vanishes (e.g., as a polynomial function with degree formula_154). 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 
formula_155 and formula_156 and formula_157
get related. This may be interpreted as arbitrarily "shifting" derivatives between formula_23 and formula_22 within the integrand, and proves useful, too (see Rodrigues' formula).
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
formula_160 and take formula_161
Begin to list in column A the function formula_162 and its subsequent derivatives formula_163 until zero is reached. Then list in column B the function formula_164 and its subsequent integrals formula_165 until the size of column B