Topics in calculus Differentiation Integration

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. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives 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 Calculus and analysis concerning the behavior of that function near a particular 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 multivariable Real analysis of vectors in an Inner In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable 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 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 has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Analysis has its beginnings in the rigorous formulation of Calculus. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The rule arises from the product rule of differentiation. In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

## The rule

Suppose f(x) and g(x) are two continuously differentiable functions. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Then the integration by parts rule states that given an interval with endpoints a, b, one has

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

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. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. Thus

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

In the traditional calculus curriculum, this rule is often stated using indefinite integrals in the form

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

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 In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative

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

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 In measure-theoretic analysis and related branches of Mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue

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

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

## Strategy

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 it into a product of two functions f(x)g(x) such that the integral produced by the integration by parts formula is easier to evaluate than the original one. The following form is useful in illustrating the best strategy to take:

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

Note that on the right-hand side, f is differentiated and g is integrated; consequently it's useful to choose f as a function that simplifies when differentiated, and/or to choose g as a function that simplifies when integrated. As a simple example, consider:

$\int \frac{\ln x}{x^2}\, dx.\!$

Since $\ln x\!$ simplifies to $1/x\!$ when differentiated, we make this part of f; since $1/x^2\!$ simplifies to $-1/x\!$ when integrated, we make this part of g. The formula now yields:

$\int \frac{\ln x}{x^2}\, dx = -\frac{\ln x}{x} - \int (1/x)(-1/x)\, dx.\!$

The remaining integral of $-1/x^2\!$ can be completed with the power rule and is $1/x\!$. This article concerns power rules for computing the Derivative in Calculus In Mathematics, the power rule is a method for differentiating

Alternatively, we may choose f and g such that the product $f'(\int g\, dx)\!$ simplifies due to cancellation. For example, suppose we wish to integrate:

$\int \frac{\ln(\sin x)}{(\cos x)^2}\, dx\!$

If we choose $f(x) = \ln(\sin x)\!$ and $g(x) = 1/(\cos x)^2\!$, then f differentiates to $1/\tan x\!$ using the chain rule and g integrates to $\tan x\!$; so the formula gives:

$\int \frac{\ln(\sin x)}{(\cos x)^2}\, dx = \ln(\sin x)\tan x - \int (1/\tan x)(\tan x)\, dx.\!$

The integrand simplifies to 1. In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. 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. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) Some other special techniques are demonstrated in the examples below.

## Examples

### Integrands with powers of x or e x

In order to calculate:

$\int x\cos (x) \, dx\!$

Let:

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

Then:

\begin{align} \int x\cos (x) \,dx & = \int u \, dv \\ & = uv - \int v \, du \\ & = x\sin (x) - \int \sin (x) \, dx \\ & = x\sin (x) + \cos (x) + C.\end{align}\!

where C is an arbitrary constant of integration. In Calculus, the Indefinite integral of a given function (ie the set of all Antiderivatives of the function is always written with a constant the constant

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 evaluated.

This example uses integration by parts twice. First let:

u = cos(x); thus du = −sin(x) dx
dv = ex dx; thus v = ex

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 = ex; dv = ex 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 (and C' = C/2) is an arbitrary constant of integration. In Calculus, the Indefinite integral of a given function (ie the set of all Antiderivatives of the function is always written with a constant the constant

A similar trick is used to find the integral of secant cubed. One of the more challenging Indefinite integrals of elementary Calculus is \int \sec^3 x \ dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan

### Interchange of the order of integration

Integration over the triangular area can be done using vertical or horizontal strips as the first step. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by The sloped line is the curve y = x.

The above formulation includes the technique of interchange of the order of integration, which is not usually viewed in this manner. Consider the double integral:

$\int_a^z \ dx\ \int_a^x \ dy \ h(y)$

In the order written above, the strip of width dx is integrated first over the y-direction as shown in the left panel of the figure, which is inconvenient especially when function h ( y ) is not easily integrated. The integral can be reduced to a single integration by reversing the order of integration as shown in the right panel of the figure. To accomplish this interchange of variables, the strip of width dy is first integrated from the line x = y to the limit x = z, and then the result is integrated from y = a to y = z, resulting in:

$\int_a^z \ dx\ \int_a^x \ dy \ h(y) = \int_a^z \ dy \ h(y)\ \int_y^z \ dx = \int_a^z \ dy \ \left(z-y\right) h(y) \ .$

This result can be seen to be an example of the above formula for integration by parts, repeated below:

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

Substitute:

$g (x) = \int_a^x \ dy \ h(y) \$  and  $f(x) = z-x \ .$

However, exchange of the order of integration has the merit that it generates the function f in a natural manner.

### More examples

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(xdx. 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 (ie the set of all Antiderivatives of the function is always written with a constant the constant

The second example is ∫ arctan(xdx, where arctan(x) is the inverse tangent function. Re-write this as

$\int \arctan (x) \cdot 1 \, dx.\!$

Now let:

u = arctan(x); du = 1/(1 + x2dx
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, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational

Here is an 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:

I: inverse trigonometric functions: arctan x, arcsec x, etc. A rule of thumb is a principle with broad application that is not intended to be strictly accurate or reliable for every situation
L: logarithmic functions: ln x, log2(x), etc. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce
A: algebraic functions: $x^2\!$, $3x^{50}\!$, etc. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations
T: trigonometric functions: sin x, tan x, etc.
E: exponential functions: $e^x\!$, $13^x\!$, etc. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x)

Then make dv the other function. You can remember the list by the mnemonic ILATE. A mnemonic device (nəˈmɒnɪk is a Memory aid Commonly met mnemonics are often verbal something such as a very short poem or a special word used to help a person remember The reason for this is that functions lower on the list have easier antiderivatives than the functions above them. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative

To demonstrate this rule, consider the integral

$\int x\cos x \, dx.\!$

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

$x\sin x - \int 1\sin x \, dx\!$

which equals

$x\sin x + \cos x+C.\!$

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

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

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

$\int x^3e^{x^2}\, dx=\frac{1}{2}e^{x^2}(x^2-1)+C\!$

## Recursive integration by parts

Integration by parts can often be applied recursively on the $\int v\,du$ term to provide the following formula

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

Here, $u'\!$ is the first derivative of $u\!$ and $u''\!$ is the second derivative of $u\!$. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition 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 ($uv\!$) differs from the previous equation. The $dv\!$ 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. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition A much easier visual representation of this process is often taught to students and is dubbed either "the tabular method," "rapid repeated integration" 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.

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 cosx) 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

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

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:

$\int e^x \cos x \,dx = e^x\sin x + e^x\cos x - \int e^x \cos x \,dx$

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

More specifically, suppose Ω is an open bounded subset of $\mathbb{R}^n$ with a piecewise smooth boundary ∂Ω. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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 piecewise-defined function (also called a piecewise function) is a function whose definition is dependent on the value of the Independent For a different notion of boundary related to Manifolds see that article If u and v are two continuously differentiable functions on the closure of Ω, then the formula for integration by parts is

$\int_{\Omega} \frac{\partial u}{\partial x_i} v \,dx = \int_{\partial\Omega} u v \, \nu_i \,d\sigma - \int_{\Omega} u \frac{\partial v}{\partial x_i} \, dx$

where $\mathbf{\nu}$ is the outward unit surface normal to ∂Ω, νi is its i-th component, and i ranges from 1 to n. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set Replacing v in the above formula with vi and summing over i gives the vector formula

$\int_{\Omega} \nabla u \cdot \mathbf{v}\, dx = \int_{\partial\Omega} u\, \mathbf{v}\cdot\nu\, d\sigma - \int_\Omega u\, \nabla\cdot \mathbf{v}\, dx$

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

Setting u equal to the constant function 1 in the above formula gives the divergence theorem. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail For $\mathbf{v}=\nabla v$ where $v\in C^2(\bar{\Omega})$, one gets

$\int_{\Omega} \nabla u \cdot \nabla v\, dx = \int_{\partial\Omega} u\, \nabla v\cdot\nu\, d\sigma - \int_\Omega u\, \Delta v\, dx$

which is the first Green's identity. In Mathematics, Green's 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 Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions In the first formula above, only $u,v\in H^1(\Omega)$ is necessary (where H1 is a Sobolev space); the other formulas have similarly relaxed requirements. In Mathematics, a Sobolev space is a Vector space of functions equipped with a norm that is a combination of ''Lp'' norms of the function

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. Stand and Deliver is a 1988 Film dramatizing the work of Jaime Escalante, a dedicated High school mathematics Teacher [1]

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