The Indefinite Integral

The new function, y — F(x) + C, which we obtain after integration, must be such that its derivative is equal to fix), to ensure that the definition conforms with the requirement that integration is the reverse (or inverse) of differentiation. Thus, we must have  

4 General Strategies for Solving More Complicated Integrals 

Some of the strategies involved in simplifying the form of an integral are quite straightforward. For example  

A chemical example of a function which does not have an explicit form can be found in thermodynamics, where the entropy is determined by integrating Cp/T. which may be known only at selected temperatures. 

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 

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Definite Integral The concept and derivation of the definite integral are completely different from those for the indefinite integral. These are by definition different types of operations. However, the formal operation J as it turns out treats the integrand in the same way for both. 

Methods of Integration All the methods of integration available for the indefinite integral can be used for definite integrals. In addition, several others are available for the latter integrals and are indicated below. 

Separation of variables and integration gives the indefinite integral, Eq. (3-8). 

The indefinite integration in dX of equation 3.1 gives the potential energy of the harmonic oscillator (cf equation 1.16) ... 

Find the indefinite integral of a number of simple functions from first principles... 

Figure 6.6 Integration of the function f(x) =18x2 (right) yields a family of functions given by the indefinite integral F(x) = 6x3+C (left), where C can take any value. Differentiation of F(x) yields the original function, f(x)...

A The first step requires us to find the indefinite integral J dx. Using the substitution u= 1 + x, the integral becomes ... 

The definition of integration as the inverse of differentiation, yielding the indefinite integral. 

Only one constant of integration will be produced from the indefinite integral J e x (x) dx and, since the constant arising from the determination of R(x) can be discarded, we see that a single constant of integration arises from the solution to a first-order differential equation, as expected. 

It follows that the functions in the left-hand column of the preceding table are the integrals of the functions in the right-hand column. More formally, they are the indefinite integrals of the function, in contrast to the definite integrals described next. Tables of indefinite integrals in reference books may be consulted for more complex examples, or mathematical software can be used to evaluate them. 

Semiintegration — Let /(f) be some time-dependent property, interest in which commences at time t = 0. The indefinite integral of this function is a well-known quantity, for which-jp-/(t) is an informative but unfa-... 

The indefinite integral, Eq. (4.16), is known as the Clausius-Clapeyron equation. Unfortunately, a plot of In p versus l/T over a significant range of l/T does not give a straight line. Consequently, Eq. (4.16) often is modified one result is the Antoine equation discussed in Sec. 3.3. A definite integral of Eq. (4.15) is... 

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