Integration by Partial Fractions

Integration by Partial Fractions is of assistance in the integration of rational fractions. If... 

The result is rather tedious to obtain, but the method can be the same as that in Example 3-5 use of the stoichiometric relationship and the introduction of , followed by integration by partial fractions and reversion to cA and cB to give... 

You have seen partial fraction solutions to integral equations in math courses. We will find many situations where rate equations can only be integrated by partial fractions so it is worth re viewing this procedure. The solutions are simple but they require some algebraic manipulation. The integrand of the previous integral can be written as... 

After integrating by partial fractions and using r(0) = 1, we find... 

The right side of Eq. (21) can be decomposed by partial fraction and integrated to give... 

The integral on the left side of Equation 5-33 can be expressed by partial fraction to give... 

These equations can be solved (although not easily) by integration by the method of partial fractions, by matrices, or by Laplace transforms. For the case where [I]o = [P]0 = 0, the concentrations are... 

In practice, the inverse Laplace transformations are obtained by reference to the rather extensive tables that are available. It is sometimes useful to develop the function in question in partial fractions, as employed in Section 3.3.3. The resulting sura of integrals can often be evaluated with the use of the tables. 

Below is an integral that is quite difficult to do by hand. It is not found in standard tables in its given form although it may transform to a recognized case. It is especially difficult to do by hand unless one notices a trick that involves performing a partial fraction decomposition of the integrand with respect to LOG(X). However, KACSYMA handles it readily. 

After integration by the method of partial fractions, the result is ... 

This integral may be evaluated either by splitting into partial fractions, or by graphical or numerical means. Using the method of partial fractions, we obtain after some fairly lengthy manipulation ... 

Integration of Equation (5.72) by the partial fraction technique yields ... 

The sharp separation of CT solids in Fig. 2 into neutral (...DADA...) and ionic (...D A"D A"...) is not satisfied by several phenazine-TCNQ complexes whose ground states are best characterized as partly-ionic, with fractional q in mixed stacks. Partial ionicity raises both theoretical and experimental questions. The main theoretical problem for weakly-overlapping sites is that any one-electron treatment, even in the Hartree-Fock limit, yields a minimum energy for integral, rather than fractional q. Perturbing the individual molecular sites is not an attractive solution for small overlap. 

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. 

In this expression, there are just sufficient equations to determine the complex system of partial fractions, by equating the coefficients of like powers of x. The integration of many of the resulting expressions usually requires the aid of one of the reduction formulae ( 76). 

Now integrate the first equation by the method of partial fractions. Evaluate the integration constant for x = 0 when t — 0 and show that... 

Any rational function of x may be integrated, if the denominator is factored into linear and irreducible quadratic factors. The function may then be broken into partial fractions, and the individual partial fractions integrated by use of the appropriate formula from the integral table. See the section on partial fractions for further information. 

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