对不定积分的新处理

NEW TREATMENT OF INDEFINITE INTEGRALS

以前用不定积分表示全体的原函数ｆ（ｘ）＋ｃ，从而把Ｃ看作任意常数或变元，比较新近的说法则用表示原函数族，从而不再是函数（而是函数集），这些说法都有其本身的问题，难以服人，本文把写成（分清现行变元与积分变元），用它表示一个确定的（但尚未具体指定的）原函数，Ｃ为待定常数，并作约定，在此基础上重新建立不定积分论．

In the classic theory the indefinite integral (t)dt orf(x)dx, the x on the sperscript is called current variable) means the general form of theprimitives of f(x), i.e. f(x) + C, C being an arbitrary constant, and F(x) being anunspecified primitive. The constant C is used in universal sense, and some books even saidthat f(x)dx was a function of two variables, x and C. Such a view-point causes muchtroublesome. For examples, the formula becomes invalid if we regard the three C's(understood in the three indefinite integrals) as in dependent variables, whether they are the same or distinct. Besides, we deduceinconsistency from'the formula of integration by part. All these are the fundamental difficulttes of the classic theory. Nowadays we ge.nerally define (x)dx as the family of theprimitive functions, and then f(x)dx is no more a function but a family (a set) of functions . This is totally different from the classic theory. In the classic, indefinite integral is afunction on which we may perform the usual (algebraic and analytic) operations. When wedefine indefinite integral as a family of functions we should discuss the corrresponding operations on families of functions. Such a discussion is, however, unnecessary and quite tedious. In this paper, we regard the indefinite integral f(x)dx as a fixed primitive, beingdetermined by specifying its zero. However such a zero needs not be given concretely beforehand. We stipulate only that the zeros should be the same when the current variablesare the same, and the zeros should correspond to the same .value-of the parameter when thecurrent variables depend on the same parameter. The C's appearing in the formulas aresuitable (but not arbitrary) constants to keep the formulas true. It is easy to verify that allformulas about indefinite integrals remain true except that a constant C shuld be added inthe formula about integration by part. Such a constant should not be omitted even in theclassic theory (otherwise we would fall into inconsistency).