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Mathematical method in calculus From Wikipedia, the free encyclopedia

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:

Or, letting and while and the formula can be written more compactly:

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.^{[1]}^{[2]} 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.

The theorem can be derived as follows. For two continuously differentiable functions and , the product rule states:

Integrating both sides with respect to ,

and noting that an indefinite integral is an antiderivative gives

where we neglect writing the constant of integration. 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 and and applying the fundamental theorem of calculus gives the definite integral version: The original integral contains the derivative v'; to apply the theorem, one must find v, the antiderivative of v', then evaluate the resulting integral

It is not necessary for and to be continuously differentiable. Integration by parts works if is absolutely continuous and the function designated is Lebesgue integrable (but not necessarily continuous).^{[3]} (If has a point of discontinuity then its antiderivative may not have a derivative at that point.)

If the interval of integration is not compact, then it is not necessary for to be absolutely continuous in the whole interval or for to be Lebesgue integrable in the interval, as a couple of examples (in which and are continuous and continuously differentiable) will show. For instance, if

is not absolutely continuous on the interval [1, ∞), 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

is not Lebesgue integrable on the interval [1, ∞), but nevertheless

with the same interpretation.

One can also easily come up with similar examples in which and 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.^{[citation needed]}

Integrating the product rule for three multiplied functions, , , , gives a similar result:

In general, for factors

which leads to

Consider a parametric curve . Assuming that the curve is locally one-to-one and integrable, 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 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.

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:

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 1/*x*, one makes (ln(*x*)) part *u*; since the antiderivative of 1/*x*^{2} is −1/*x*, one makes 1/*x*^{2} part *v*. The formula now yields:

The antiderivative of −1/*x*^{2} can be found with the power rule and is 1/*x*.

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^{2}x, then *u* differentiates to using the chain rule 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, 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.

In order to calculate

let:

then:

where *C* is a constant of integration.

For higher powers of in the form

repeatedly using integration by parts can evaluate integrals such as these; each application of the theorem lowers the power of by one.

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 (and ) is a constant of integration.

A similar method is used to find the integral of secant cubed.

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 is also known.

The first example is . We write this as:

Let:

then:

where is the constant of integration.

The second example is the inverse tangent function :

Rewrite this as

Now let:

then

using a combination of the inverse chain rule method and the natural logarithm integral condition.

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:^{[4]}

**L**– logarithmic functions: etc.**I**– inverse trigonometric functions (including hyperbolic analogues): etc.**A**– algebraic functions (such as polynomials): etc.**T**– trigonometric functions (including hyperbolic analogues): etc.**E**– exponential functions: 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 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

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.

The Wallis infinite product for

The gamma function is an example of a special function, defined as an improper integral 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:

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.

If is a -times continuously differentiable function and all derivatives up to the th one decay to zero at infinity, then its Fourier transform satisfies

where is the th derivative of . (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 . A similar method can be used to find the Laplace transform of a derivative of a function.

The above result tells us about the decay of the Fourier transform, since it follows that if and are integrable then

In other words, if satisfies these conditions then its Fourier transform decays at infinity at least as quickly as 1/|*ξ*|^{k}. 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 1 + |2π*ξ*^{k}| gives the stated inequality.

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 (see *L*^{p} space). If is smooth and compactly supported then, using integration by parts, we have

- Determining boundary conditions in Sturm–Liouville theory
- Deriving the Euler–Lagrange equation in the calculus of variations

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 n 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 nth 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