The Antiderivative of a Function

In Chapter 4, we discussed the derivative of a function. We consider the inverse problem, finding a function that possesses a specific function as its derivative. We begin with a particular example. 

The position of a particle is represented by its position vector, which we denote by r. If a particle moves only in the vertical direction, we can e q)ress its position as a function of time by the z component of this vector, which is a function of time. 

The velocity is the derivative of the position vector with respect to time. The velocity is a vector which we denote by v. The z component of the velocity is 

The X and y components are defined in the same way if the particle moves in three dimensions. 

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The antiderivative of a function fix) will hereafter be called the indefinite integral and be designated J fix) dx. Thus, the result derived in the last paragraph... 

We define the indefinite integral (or antiderivative) of a function f x) as the function F x) that, when differentiated, yields the original function/(x). That is,... 

The integral on the left can be solved by using the double angle formula for sinh X, then multiplying top and bottom by cosh ( /4) and integrating by an appropriate substitution. This works because the substitutions make it clear that we are integrating the ratio of a function to its antiderivative, a common pattern. 

The fundamental difficulty in solving DEs explicitly via finite formulas is tied to the fact that antiderivatives are known for only very few functions / M —> M. One can always differentiate (via the product, quotient, or chain rule) an explicitly given function f(x) quite easily, but finding an antiderivative function F with F x) = f(x) is impossible for all except very few functions /. Numerical approximations of antiderivatives can, however, be found in the form of a table of values (rather than a functional expression) numerically by a multitude of integration methods such as collected in the ode... m file suite inside MATLAB. Some of these numerical methods have been used for several centuries, while the algorithms for stiff DEs are just a few decades old. These codes are... 

To take full advantage of the group-theoretical formulation, it is desirable to write the pre-factor 2k/ ih) in eq.(28) as a group integral of a suitable operator. This is achieved introducing the antiderivative operator A [12], which acts on functions f(s) L (R), Lf(R) being the linear subspace of Li(R) such that... 

Equation 3.6 implies that U(r) is the negative of the antiderivative of F(r), so Equation 3.6 does not uniquely define U(r). A different potential energy function V(r ) = U(r) + C, where C is any numerical constant, would give the same force ... 

Now consider the reverse problem from that of the previous example. If we are given the acceleration as a function of time, how do we find the velocity If we are given the velocity as a function of time, how do we find the position In the following example, we see that the answer to these questions involves the antiderivative function, which is a function that possesses a particular derivative. 

F(l) then describes the slope of the graphical representation of the function W(l) at point 1. In order to find W 1), we need only to find the antiderivative of F 1). In this case, it is the function whose derivative with respect to I results in D 1 — Iq). Antiderivatives are gone into in more detail in Sect. A.1.3 in the Appendix. Now we see that ... 

Next, a tail model corresponding to the antiderivative of the trial expression for g r) was fitted to the truncated numerical integral of g(r) as a function of the upper integration limit. This tail model was then used to extrapolate HiRy to = >, which yielded the value of the TCFI. Defining the running integrals of the RDF, Gif),... 

The velocity is the antiderivative of the given acceleration function plus a constant vq ... 

Therefore, solving a DE by obtaining a table of approximate values for the antiderivative F of a given function / is a cinch nowadays. While theoretical studies of DEs helps us to understand them, it does not help with actually solving DEs for applied problems. 

Usually, however, we would prefer to have an explicit functional form for the integral. Since integration is the inverse operation of differentiation, this means that to integrate /(x) we need to find a function whose derivative is fix). This new function is called the antiderivative. The difference between the values of this antiderivative function at the two extreme limits of the area gives the value of the integral. For example, we already showed (Equation 2.3) that the derivative of x3 is 3x2, so x3 is an antiderivative of 3x2. Thus we have ... 

In the previous examples we have identified an antiderivative function by inspection of the well-known formulas for derivatives. We now consider the general problem of constructing a function that possesses a certain derivative. Say that we have a function / = f(x), and we want to find its antiderivative function, which we call Fix). That is,... 

Equation (5.22) is an important equation. In many applications of calculus to physical chemistry, we will be faced with an integral equivalent to the right-hand side of this equation. If we can by inspection or by use of a table find the function F that possesses the function / as its derivative, we can evaluate the function F at the two limits of integration and take the difference to obtain the value of the integral. In other cases, we might not be able to identify the antiderivative function, but can numerically construct a change in its value using this equation. 

This integral is called an indefinite integral, since the lower limit is unspecified and the upper limit is variable. The indefinite integral is the same as the antiderivative function. Large tables of indefinite integrals have been compiled. Appendix E is a brief version of such a table. In most such tables, the notation of Eq. (5.33) is not maintained. The entries are written in the form... 

Since the indefinite integral is the antiderivative function, it is used to find a definite integral in the same way as in Section 5.2. If x and X2 are the limits of the definite integral,... 

We obtain first examples of antiderivatives by simply reversing the arrows in the expressions (A.6) to (A.8). F x) = Inr is then an antiderivative of fix) = 1/r. However, it is only possible to determine an antiderivative up to a constant additive term C because it drops out when taking the derivative. Along with F x), aU functions F(x) + C are an antiderivative of/(x). 

Let us consider a problem that, at first glance, does not appear to have anything to do with finding an antiderivative, namely determining the area under a curve of an arbitrary function y =f(x). This is the area that is bordered by the curve between two limits Xi and X2 and the x-axis (Fig. A.6). We obtain approximations of surface area A by dividing the surface into strips having width Ax, each one limited by a horizontal line at its function value and then adding the areas/(x) Ax of these strips ... 

Further, Since p is continuous for a < x < /3, it follows that p is defined in this interval and is a nonzero differentiable function. Thus, the conversirai of Equation 2.29 into the form of Equation 2.32 is justified. Also, the functitm //.ghas an antiderivative because p and g are ccmtinuous and Equation 2.33 follows from Equation 2.32. The assumption that there is at least one solution of EquatiOTi 229 is verifiable by substituting Equation 2.33 into Equation 2.29. The initial condition. Equation 2.30, determines the integratimi cmistant c uniquely. 

The left-hand side of this equation contains a conventional symbol for a definite integral a vertical line segment following the antiderivative function with the lower limit written at the bottom and the upper limit written at the top. Equation (7.14) is often called tht fundamental theorem of integral calculus. The antiderivative function F is called the indefinite integral of the integrand function /. 

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