NationMaster - Encyclopedia: Integration by parts

Topics in calculus
Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem For other uses, see Calculus (disambiguation). ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...
Differentiation
Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives For a non-technical overview of the subject, see Calculus. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable. ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ... The primary operation in differential calculus is finding a derivative. ...
Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution This article is about the concept of integrals in calculus. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. The rule arises from the product rule of differentiation. For other uses, see Calculus (disambiguation). ... Analysis has its beginnings in the rigorous formulation of calculus. ... This article is about the concept of integrals in calculus. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... For a non-technical overview of the subject, see Calculus. ...

1 The rule
2 Examples
3 The ILATE rule
4 Recursive integration by parts
5 Tabular integration by parts
6 Higher dimensions
7 Cultural references
8 References
9 External links

The rule

Suppose f(x) and g(x) are two continuously differentiable functions. Then the integration by parts rule states that given an interval with endpoints a, b, one has In mathematics, a smooth function is one that is infinitely differentiable, i. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

$int_a^b f(x) g'(x),dx = left[ f(x) g(x) right]_{a}^{b} - int_a^b f'(x) g(x),dx$

where we use the common notation

$left[ f(x) g(x) right]_{a}^{b} = f(b) g(b) - f(a) g(a).$

The rule is shown to be true by using the product rule for derivatives and the fundamental theorem of calculus. Thus In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ...

$f(b)g(b) - f(a)g(a),$	$= int_a^b frac{d}{dx} ( f(x) g(x) ) , dx$
	$=int_a^b f'(x) g(x) , dx + int_a^b f(x) g'(x) , dx.$

In the traditional calculus curriculum, this rule is often stated using indefinite integrals in the form In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

$int f(x) g'(x),dx = f(x) g(x) - int f'(x) g(x),dx,$

or in an even shorter form, if we let u = f(x), v = g(x) and the differentials du = f ′(x) dx and dv = g′(x) dx, then it is in the form in which it is most often seen:

$int u,dv=uv-int v,du.$

Note that the original integral contains the derivative of g; in order to be able to apply the rule, the antiderivative g must be found, and then the resulting integral ∫g f′ dx must be evaluated. In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

One can also formulate a discrete analogue for sequences, 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. ...

An alternative notation has the advantage that the factors of the original expression are identified as f and g, but the drawback of a nested integral:

$int f g,dx = f int g,dx - int left ( f' int g,dx right ),dx.$

This formula is valid whenever f is continuously differentiable and g is continuous.

More general formulations of integration by parts exist for the Riemann-Stieltjes integral and Lebesgue-Stieltjes integral. In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ... If you are having difficulty understanding this article, you might want to first learn more about integrals, particularly the Lebesgue integral, and measure theory. ...

Note: More complicated forms such as the one below are also valid:

$int u v,dw = u v w - int u w,dv - int v w,du.$

Examples

In order to calculate:

$int xcos (x) ,dx$

Let:

u = x, so that du = dx,

dv = cos(x) dx, so that v = sin(x).

Then:

$begin{align} int xcos (x) ,dx & = int u ,dv & = uv - int v ,du & = xsin (x) - int sin (x) ,dx & = xsin (x) + cos (x) + C end{align}$

where C is an arbitrary constant of integration. In calculus, the indefinite integral of a given function (i. ...

By repeatedly using integration by parts, integrals such as

$int x^{3} sin (x) ,dx quad mbox{and} quad int x^{2} e^{x} ,dx$

can be computed in the same fashion: each application of the rule lowers the power of x by one.

An interesting example that is commonly seen is:

$int e^{x} cos (x) ,dx$

where, strangely enough, in the end, the actual integration does not need to be performed.

This example uses integration by parts twice. First let:

u = cos(x); thus du = −sin(x) dx

dv = e^x dx; thus v = e^x

Then:

$int e^{x} cos (x) ,dx = e^{x} cos (x) + int e^{x} sin (x) ,dx$

Now, to evaluate the remaining integral, we use integration by parts again, with:

u = sin(x); du = cos(x) dx

v = e^x; dv = e^x dx

Then:

$int e^{x} sin (x) ,dx$

$= e^{x} sin (x) - int e^{x} cos (x) ,dx$

Putting these together, we get

$int e^{x} cos (x) ,dx = e^{x} cos (x) + e^x sin (x) - int e^{x} cos (x) ,dx$

Notice that the same integral shows up on both sides of this equation. So we can simply add the integral to both sides to get:

$2 int e^{x} cos (x) ,dx = e^{x} ( sin (x) + cos (x) ) + C$

$int e^{x} cos (x) ,dx = {e^{x} ( sin (x) + cos (x) ) over 2} + C$

where, again, C is an arbitrary constant of integration. In calculus, the indefinite integral of a given function (i. ...

A similar trick is used to find the integral of secant cubed. One of the more challenging indefinite integrals of elementary calculus is This antiderivative may be found by integration by parts, as follows: where Then Next we add to both sides of the equality just derived: Then divide both sides by 2. ...

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) dx. We write this as:

$int ln (x) cdot 1 ,dx$

Let:

u = ln(x); du = 1/x dx

v = x; dv = 1·dx

Then:

$int ln (x) ,dx$	$= x ln (x) - int frac{x}{x} ,dx$
	$= x ln (x) - int 1 ,dx$

$int ln (x) ,dx = x ln (x) - {x} + {C}$

$int ln (x) ,dx = x ( ln (x) - 1 ) + C$

where, again, C is the arbitrary constant of integration In calculus, the indefinite integral of a given function (i. ...

The second example is ∫ arctan(x) dx, where arctan(x) is the inverse tangent function. Re-write this as: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

$int arctan (x) cdot 1 ,dx$

Now let:

u = arctan(x); du = 1/(1+x²) dx

v = x; dv = 1·dx

Then:

$int arctan (x) ,dx$	$= x arctan (x) - int frac{x}{1 + x^2} ,dx$
	$= x arctan (x) - {1 over 2} ln left( 1 + x^2 right) + C$

using a combination of the inverse chain rule method and the natural logarithm integral condition. In calculus, the inverse chain rule is a method of integrating a function which relies on guessing the integral of that function, and then differentiating back using the chain rule. ... The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ...

Here is a fun example:

$int x dx = x^2 - int x dx$

$2 int x dx = x^2$

$int x dx = frac{x^2}{2} + C$

The ILATE rule

A rule of thumb for choosing which of two functions is to be u and which is to be dv is to choose u by whichever function comes first in this list: A rule of thumb is an easily learned and easily applied procedure for approximately calculating or recalling some value, or for making some determination. ...

I: inverse trigonometric functions: arctan x, arcsec x, etc.

L: logarithmic functions: ln x,

log 2 (x)

, etc.

A: algebraic functions:

x 2

3 x 50

, etc.

T: trigonometric functions: sin x, tan x, etc.

E: exponential functions:

e x

13 x

, etc.

Then make dv the other function. You can remember the list by the mnemonic ILATE. The reason for this is that functions lower on the list have easier antiderivatives than the functions above them. In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... In mathematics, if two variables of bn = x are known, the third can be found. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The exponential function is one of the most important functions in mathematics. ... For other uses, see Mnemonic (disambiguation). ... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

To demonstrate this rule, consider the integral

$int xcos x ,dx.,$

Following the ILATE rule, u = x and dv = cos x dx , hence du = dx and v = sin x , which makes the integral become

$xsin x - int 1sin x ,dx,,$

which equals

$xsin x + cos x+C. ,$

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 as dv, we would have the integral

$frac{x^2}2cos x + int frac{x^2}2sin x,dx,,$

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 ILATE rule. A common alternative is to consider the rules in the "LIATE" order instead. Also, in some cases, polynomial terms need to be split in non-trivial ways. For example, to integrate

$int x^3e^{x^2},dx,$

we would set

$u=x^2,quad dv=xe^{x^2},dx.$

This results in

$int x^3e^{x^2},dx=frac12e^{x^2}(x^2-1)+C.$

Recursive integration by parts

Integration by parts can often be applied recursively on the $int v,du$ term to provide the following formula This article is about the concept of recursion. ...

$int uv = u v_1 - u' v_2 + u'' v_3 - cdots + (-1)^{n} u^{(n)} v_{n+1}$

Here, $u'$ is the first derivative of $u$ and $u''$ is the second derivative of $u$ . Further, $u (n)$ is a notation to describe its nth derivative (with respect to the variable u and v are functions of). Another notation has been adopted:

$v_{n+1}(x)=int! int cdots int v (dx)^{n+1}.$

There are n + 1 integrals.

Note that the integrand above ( $u v$ ) differs from the previous equation. The $d v$ factor has been written as $v$ purely for convenience.

The above mentioned form is convenient because it can be evaluated by differentiating the first term and integrating the second (with a sign reversal each time), starting out with $u v 1$ . It is very useful especially in cases when $u (k + 1)$ becomes zero for some k + 1. Hence, the integral evaluation can stop once the $u (k)$ term has been reached.

Tabular integration by parts

While the aforementioned recursive definition is correct, it is often tedious to remember and implement. A much easier visual representation of this process is often taught to students and is dubbed either "the tabular method" or "the tic-tac-toe method". This method works best when one of the two functions in the product is a polynomial, that is, after differentiating it several times one obtains zero. It may also be extended to work for functions that will repeat themselves. This article is about the concept of recursion. ...

For example, consider the integral

$int x^3 cos x ,dx.$

Let $u = x 3 .$ Begin with this function and list in a column all the subsequent derivatives until zero is reached. Secondly, begin with the function v (in this case $cos x$ ) and list each integral of v until the size of the column is the same as that of u. The result should appear as follows.

Derivatives of u (Column A)	Integrals of v (Column B)
$x^3 ,$	$cos x ,$
$3x^2 ,$	$sin x ,$
$6x ,$	$-cos x ,$
$6 ,$	$-sin x ,$
$0 ,$	$cos x ,$

Now simply pair 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 alternating signs (beginning with the positive sign). Do so until further pairing is impossible. The result is the following (notice the alternating signs in each term):

$(+)(x^3)(sin x) + (-)(3x^2)(-cos x) + (+)(6x)(-sin x) + (-)(6)(cos x) + C ,.$

Which, with simplification, leads to the result

$x^3sin x + 3x^2cos x - 6xsin x - 6cos x + C. ,$

With proper understanding of the tabular method, it can be extended.

$int e^x cos x ,dx.$

Derivatives of u (Column A)	Integrals of v (Column B)
$e^x ,$	$cos x ,$
$e^x ,$	$sin x ,$
$e^x ,$	$-cos x ,$

In this case in the last step it is necessary to integrate the product of the two bottom cells obtaining:

$int e^x cos x ,dx = e^xsin x + e^xcos x - int e^x cos x ,dx$

Which is then solvable in the usual way.

Higher dimensions

The formula for integration by parts can be extended to functions of several variables. Instead of an interval one needs to integrate over a n-dimensional set. Also, one replaces the derivative with a partial derivative. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...

More specifically, suppose Ω is an open bounded subset of $mathbb{R}^n$ with a piecewise smooth boundary ∂Ω. If u and v are two continuously differentiable functions on the closure of Ω, then the formula for integration by parts is In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...

$int_{Omega} frac{partial u}{partial x_i} v ,dx = int_{partialOmega} u v , nu_i ,dsigma - int_{Omega} u frac{partial v}{partial x_i} , dx$

where $mathbf{nu}$ is the outward unit surface normal to ∂Ω, ν_i is its i-th component, and i ranges from 1 to n. Replacing v in the above formula with v_i and summing over i gives the vector formula A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

$int_{Omega} nabla u cdot mathbf{v}, dx = int_{partialOmega} u, mathbf{v}cdotnu, dsigma - int_Omega u, nablacdot mathbf{v}, dx$

where v is a vector-valued function with components v₁, ..., v_n.

Setting u equal to the constant function 1 in the above formula gives the divergence theorem. For $mathbf{v}=nabla v$ where $vin C^2(bar{Omega})$ , one gets In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ...

$int_{Omega} nabla u cdot nabla v, dx = int_{partialOmega} u, nabla vcdotnu, dsigma - int_Omega u, Delta v, dx$

which is the first Green's identity. Greens identities are a set of three identities in vector calculus. ...

The regularity requirements of the theorem can be relaxed. For instance, the boundary ∂Ω need only be Lipschitz continuous. In the first formula above, only $u,vin H^1(Omega)$ is necessary (where H¹ is a Sobolev space); the other formulas have similarly relaxed requirements. In ordinary English, regular is an adjective or noun used to mean in accordance with the usual customs, conventions, or rules, or frequent, periodic, or symmetric. ... In mathematics, a function f : M → N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) ≤ K d(x, y) for all x and y... In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivatives up to some order k the condition of finite Lp norm, for given p â‰¥ 1. ...

For reference, consult Appendix C of Evans or the applied math notes of Arbogast and Bona.

Cultural references

The method of tabular integration by parts is featured in the 1988 film Stand and Deliver.^[1]

For other uses, see Stand and Deliver (disambiguation). ...

References

Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2.
Arbogast, Todd; Jerry Bona (2005). Methods of Applied Mathematics.
Horowitz, David (September 1990). "Tabular Integration by Parts". The College Mathematics Journal 21 (4): 307-311.

^ Horowitz, David (September 1990). "Tabular integration by parts". The College Mathematics Journal 21: 307–311.

External links

Integration by Parts - From MathWorld
Tabular Integration by Parts
Tabular Integration by Parts Demonstrated

Wikibooks' [[wikibooks:|]] has more about this subject:

Calculus/Integration techniques

Category: Integral calculus Image File history File links Wikibooks-logo-en. ...

Results from FactBites:

Integration by parts (547 words)
	In mathematics, integration by parts is a general rule that transforms the integral of calculus of products of functions into other integrals.
	The other two famous examples are when you take something which isn't a product as a product of 1 and itself, and use integration by parts.
	Integration by parts follows from the product rule of differentiation: If the two continuously differentiable functions u(x) and v(x) are given, the product rule states that

Integration by Parts (160 words)
	When using this formula to integrate, we say we are "integrating by parts".
	We can also sometimes use integration by parts when we want to integrate a function that cannot be split into the product of two things.
	To integrate this, we use a trick, rewrite the integrand (the expression we are integrating) as 1.lnx.

More results at FactBites »

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Contents

The rule GA_googleFillSlot("encyclopedia_square");

Examples

The ILATE rule

Recursive integration by parts

Tabular integration by parts

Higher dimensions

Cultural references

References

External links

The rule