Integration by substitution

A very important example of integration by substitution, is that of the function of Gauss, exp(-z2), which is shown in Fig. 4. In practical applications this function can be written in the form... 

The right-hand side of Eq. (2-17) can be reduced to an exponential integral by substituting y = E/RT to give... 

I the sum of the kinetic and potential energy of an electron in the orbital lUg in the electro-atic field of the two bare nuclei. This integral can in turn be expanded by substituting the... 

Since the form of the particular integral is known, the constants may be evaluated by substitution in the differential equation. 

The variables are separable, but an integration in closed form is not possible because of the odd exponent. Numerical integration followed by substitution into (4) will provide both A and B as functions of t. The plots, however, are of solutions of the original differential equations with ODE. 

When one of these three conditions is apphcable, the appropriate design equation can be obtained by substitution into Eq. (14-71), followed by integration of the resulting relationship. 

Integrating Equation 5-277 by substituting Cg/C and subsequent manipulation yields... 

Discretizating by substituting the various finite-difference type approximations for the terms in the integrated equation representing flow processes, which converts the integral equations into a system of algebraic equations. 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

The CNDO method has been modified by substitution of semiempirical Coulomb integrals similar to those used in the Pariser-Parr-Pople method, and by the introduction of a new empirical parameter to differentiate resonance integrals between a orbitals and tt orbitals. The CNDO method with this change in parameterization is extended to the calculation of electronic spectra and applied to the isoelectronic compounds benzene, pyridine, pyri-dazine, pyrimidine and pyrazine. The results obtained were refined by a limited Cl calculation, and compared with the best available experimental data. It was found that the agreement was quite satisfactory for both the n TT and n tt singlet transitions. The relative energies of the tt and the lone pair orbitals in pyridine and the diazines are compared and an explanation proposed for the observed orders. Also, the nature of the lone pairs in these compounds is discussed. 

Duhem s equation was integrated by Margules (Sitzungsber. Wien Akad.j 1895) by means of the substitutions ... 

Solution This can be done by substituting the various rate equations into Equation (1.36), integrating, and applying the initial condition of Equation (1.37). Two versions of these equations can be used for a second-order reaction with two reactants. Another way is to use the previous results for... 

Laplace transforms 279-286 mapping of a function 271 integrating factor 56-59, 86, 360 integration 43-61 along a curve 51-52 by parts 46 by substitution 45-46 chemical kinetics 47-48 constant of integration 43,49, 89, 124... 

Integration of Eq. (74) with respect to V from zero to infinity, followed by substitution of Eq. (75) results in an explicit expression for the rate of change of the pellet population, as follows ... 

The one-electron integral in Eq. (2.19) can then be evaluated by substitution of the MO expressions obtained from the relevant Slater determinant [Eq. (2.15)] and the application of the requirements for MO orthonormality in the resulting integral expressions [Eq. (2.3)]. After a substantial amount of algebraic manipulation, it can be shown that the relevant integral can be expressed as the sum of simple one-electron integrals ... 

On setting Es = TEr and expanding equation 46-82 out by substituting the limits of integration ... 

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