Integration by parts method

The second integration technique, known as the substitution method, derives from the inversion of the chain rule for differentiation described in Chapter 4. The objective here, once again, is to transform the integrand into a simpler or, preferably, a standard form. However, just like the integration by parts method, there is usually a choice of substitutions and although, in some cases, different substitutions yield different answers, these answers must only differ by a constant (remember that, for an indefinite integral, the answer is determined by inclusion of a constant). The substitution method is best illustrated using a worked problem. 

The use of integration by parts method for integrating products of functions. 

The integral of sin ( 0) can be carried out using the integration by parts method (see The Chemistry Maths Book from the Further Reading section in this appendix). 

The integration by parts method can be used to simplify the first term on the LHS of the problem Eq. (12.36) ... 

Apply the integration by parts and substitution methods to integrate more complicated functions... 

In practice, we may find ourselves faced with more complicated functions, the solutions to which require us to use methods involving adaptation of some of the rules for differentiation. The choice of method more often than not involves some guesswork, but coming up with the correct guesses is all part of the fun In addition, it may be necessary to use a combination of several methods. In the following two sections, we discuss integration by parts and the substitution method... 

Using the methods developed in Chapter 5, one rids the expression of variations in 4,- using an integration by parts... 

Of course, there are a number of manipulations (integration by parts, for example) that can be carried out in order to restate this result in a more convenient fashion. On the other hand, our intention at this point is to introduce the conceptual framework, and to reserve the explicit use of the method of eigenstrains until later. 

The favourite methods for integration are by processes known as the substitution of a new variable/ integration by parts and by resolution into partial fractions. The student is advised to pay particular attention to these operations. Before proceeding to the description of these methods, I will return once more to the integration constant. 

A complex integral can often be reduced to one of the standard forms by the method of integration by parts . By a repeated application of this method, complicated expressions may often be integrated, or else, if the expression cannot be integrated, the non-integrable part may be reduced to its simplest form. This procedure is sometimes called integration by successive reduction. See Ex. (5), above. 

Formulae B, C, D, E have been deduced from (1), page 208, by the method of integration by parts. Perhaps the student can do this for himself. The reader will notice that formula B decreases (algebraically) the exponent of the monomial factor from m to m - n + 1, while C increases the exponent of the same factor from m to m +1. Formula D decreases the exponent of the binomial factor from p iop - 1, while E increases the exponent of the binomial factor from p to p + 1. B and D fail when np + m + 1 = 0 C fails when m + 1 = 0 E fails when p + 1 = 0. When B, C, and D fail use F, page 204 if E fails p = - 1 and the preceding methods apply. 

The Galerkin method is applied with linear finite elements. The weak formulation of Eqs. (la) and (lb) is obtained by taking simultaneously the products of the equations with appropriate test functions and integration by parts of the spatial derivatives. We use a Lagrangian interpolation of the approximate solutions C for the aqueous solute concentration C, and S for the sorbed phase concentration Si for every species ... 

Integration by parts is another method suggested by the formula for the derivative of a product (Eq. 6.45). In differential form, this can be expressed as... 

Using the method of integration by parts, we can rewrite the term involving in the following manner,... 

For systems with nonlinearities of polynomial form another method to calculate statistical properties of the response process is based on the Fokker-Planck equation If eq. (8) is multiplied by Xj vx2 2-... Xn n where r, are non-negative integers - and integrated by parts over the whole domain of definition, a set of differential equations in terms of moments is derived that is linear, but not closed. A closure can be achieved by... 

Table A9.4 The first few integrals of the type required for the integration of the radial functions. The n = 0 case is straightforward and the other solutions are obtained by repeated application of the method of integration by parts. Note with the limits shown the result Is l = n/p". ...

Trigonometric Functions. In modeling processes and in studying control systems, there are many other important time functions, such as the trigonometric functions, cos cor and sin cor, where co is the frequency in radians per unit time. The Laplace transform of cos cor or sin cor can be calculated using integration by parts. An alternative method is to use the Euler identity ... 

O Equation 26.35 can be integrated by parts in order to shift the differential operator from the field variable u to the weighting function w. Depending on the order of integration and the choice of the weighting function w, it is possible to derive based on the weighted residual method all the classical approximation methods, cf O Fig. 26.15 and (Brebbia et al. 1984 Zienkiewicz and Taylor 2000). 

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