Antiderivative

In indefinite integration (also knows as antidifferentiation), there is extra information in the constant of integration. It is, strictly speaking, incorrect to say that "the antiderivative of x is because is only one of many antiderivatives, including x + 1.7 x 10 But the statement is correct in spirit the difference between x and any antiderivative of x is irrelevant for most purposes. Equivalence classes are the mathematician s way to make precise the notion of irrelevant ambiguity. 

The expression 5/ is often pronounced 5 modulo equivalence or S mod equivalence. If possible and convenient, we refer to the equivalence by name for example, vectors are directed line segments modulo translation and antiderivatives are functions modulo constants . We leave the details of applying Definition 1.3 to vectors and antiderivatives to the interested reader in Exercises 1.19 and 1.20. 

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... 

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. 

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... 

To find the explicit expression of the r.h.s. in eq.(33) we have to show how the antiderivative operator acts. In this case we have... 

This result concludes the presentation of the Heisenberg group approach as the powerful tool that allows to derive classical mechanics as a formal limit of quantum mechanics, for h —> 0. The most important ingredients that have been introduced to obtain this result are the Fourier-like representation of observables and equations of motion and the definition of the antiderivative operator. These elements will be used in section 5 to derive a similiar procedure for a mixed quantum-classical mechanics. An ansatz on the quantum-classical equations of motion will be necessary, but the subsequent application of Heisenberg group formalism will be a straightforward generalization of what has been done so far. 

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 ... 

Since the derivative of any constant is zero, the antiderivative can only be determined up to an added constant. For example, the functions fix) =x3,/(x) =x3 + 12, and fix) = x3 — 3 all have the same derivative idfix)/dx = 3x 2). But if you redo the integration in Equation 2.26 using either of these other functions as the antiderivative, you end up with the same answer for the integral. This means that the additive constant can be chosen to be whatever value is convenient, which will be quite important when we consider potential energy in Chapter 3. 

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 ... 

F is called the antiderivative of f because, for reasons we shall not go into here, it is obtained by the inverse of a derivative operation. In other words, if F is the anti-derivative of f, then / is the derivative of F ... 

Finally, if the upper and lower limits of integration are reversed, the sign of the integral changes as well. This is easily seen from the antiderivative form ... 

Noticing that the function g x, y) is the antiderivative of the integrand on the right side of the previous expression, rue have... 

An indefinite integral is the same thing as the antiderivative function. 

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. 

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. 

SOLUTION From our table of derivatives we find that the antiderivative of sin (x) is... 

The constant C cancels because it occurs with the same value in both occurrences of the antiderivative function. ... 

When we found the value of the integral from the antiderivative function, we omitted the constant term that generally must be present in the antiderivative function. This constant would have canceled out, so we left it out from the beginning. 

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... 

The same information is contained in a table of indefinite integrals as is contained in a table of derivatives. However, we can get by with a fairly short table of derivatives, since we have the chain rule and other facts listed in Section 4.4. Antiderivatives are harder to find, so it is good to have a separate table of indefinite integrals, arranged so that similar integrand functions occur together. 

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,... 

The antiderivative of either side of this equation is just uv + C, where C is an arbitrary constant. We can write the indefinite integral... 

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