Table of Integrals

This is also a gamma function and may be solved with the help of a table of integrals. Evaluation of the integral gives the simple result... 

This integral is a gamma function and is readily solved using a table of integrals. In writing the last result uIq has been replaced by rQ the mean-square coil dimensions under 0 conditions. Equation (9.49) involves rjj" not r so we note that rQ = nlQ and replace n by I i - j I, the number of units separating units i and j, to obtain... 

The rate equations will be stated in these terms for a number of reactions. In all these cases, the integrands are ratios of second-degree equations. The moderately complex integrations are accomplished with the aid of a table of integrals, or by MATHEMATICA, or numerically when the constants are known. 

Equation 9-15 gives the conversion expression for the second order reaction of a macrofluid in a mixed flow. An exponential integral, ei(a), which is a function of a, and its value can be found from tables of integrals. However, the conversion from Equation 9-15 is different from that of a perfectly mixed reactor without reference to RTD. An earlier analysis in Chapter 5 gives... 

S. Gtadshtein and 1 Jvl. Ryzhik, Table of integrals, series and product , (Academic Press, San Diego, 1980) 16. L.W. Johnson and RD. Riess, Numerical Analysis , 2nd edition, (Addison Wesley, London, 1982)... 

Integration by Tables is possible if an integral may be put into a form that can be found in a table of integrals, such as the one given in Table 1-7. More complete tables may be found in Bois, Tables of Indefinite Integrals, Dover, and in others. 

See any standard text, such as Erdelyi, Magnus, Oberhettmger, and Tricomi, Tables of Integral Transforms, McGraw-Hill Book Co., New York, 1954. 

Dwight, H.B. Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1957). 

S. Gradshteyn and 1. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1980. 

B. 0. Peirce, A Short Table of Integrals, Gina and Company, Boston, 1929. 

REFERENCES de Brujin, N. G. Asymptotic Methods in Analysis, Dover, New York (1981) Folland, G. B., Advanced Calculus, Prentice-Hall, Saddle River, N.J. (2002) Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, Series, and Troducts, Academic, New York (2000) Kaplan, W., Advanced Calculus, 5th ed., Addison-Wesley, Redwood City, Calif. (2003). 

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals Series and Products, 4th ed., Academic Press New York (1965). 

This is the conversion expression for second-order reaction of a macrofluid in a mixed flow reactor. The integral, represented by ei(a) is called an exponential integral. It is a function alone of a, and its value is tabulated in a number of tables of integrals. Table 16.1 presents a very abbreviated set of values for both ei(jc) and Ei(jc). We will refer to this table later in the book. 

From the table of integrals in Table 16.1 we find ei(2) = 0.048 90 from which Micro-late, and macro-late or early C" = 0.362... 

Integrating by parts using the recursion formula, found in any table of integrals, we obtain... 

Gradshteyn, I. S., and Ryzhik, I. M., Table of Integrals, Series, and Products, 5th ed., Academic Press, New York, 1993. (There are numerous excellent reference books available on mathematical formulas. This book has a reasonably large collection, far more than what is needed for the present book. A CD-ROM version of this book is also available CD-ROM Version 1, Alan Jeffrey, Ed.)... 

A third method—solution by Laplace transforms—can be used to derive many of the results already mentioned. It is a powerful method, particularly for complicated problems or those with time-dependent boundary conditions. The difficult part of using the Laplace transform is back-transforming to the desired solution, which usually involves integration on the complex domain. Fortunately, Laplace transform tables and tables of integrals can be used for many problems (Table 5.3). 

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