Advanced Calculus - download pdf or read online

By Wilson E.B.

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Jy It frequently turns out that the integrals on the right are easily obtained by methods already given; otherwise they can be treated by the substitutions above. In addition to these substitutions there are a large number of others which are Many of them will be found among the exerMoreover, it frequently happens that an integrand, which does not come under any of the standard types for which substitutions are indicated, is none the applied under specific conditions. cises. less integrable by some substitution which the form of the integrand will suggest.

This theorem And in reality nothing bub a restatement of the definition of conti- is u nuity combined with the definition of a limit. For lim/(x) =/(a) when x = a, no matter how" means that the difference between /(x) and /(a) can be made as small as desired by taking x and conversely. The reason more amenable to analytic opera- near to a sufficiently for this restatement is that the present form is ; albo suggests the geometric picture which corresponds to the usual idea of continuity in graphs.

Several of these reduction formulas are cited among the exercises. Although the Table is useful in performing integrations and indeed makes it to a, large extent unneceslearn to the various methods of sary integration, the exercises immediately below, which are constructed for the purpose of illustrating methods of integration, should be done without the aid of a Table. EXERCISES 1. Integrate the following (a) J*x cosh by parts : 1 (|8) , J'tan- z, (7) Cx m log *, ~ sin la; 2. If P(x) is () (/3) and fP a polynomial and P'(K), (x) e* = e*[p Jp (x) cos ax = (x) - sin ax P"(cc), its derivatives, show - I P'(x) + 1 P"(x) ---- 1 |P (x) (7) derive a similar result for the - -^ P"(x] + ~ P*(o;) ---- 1 integrand P (x) sin ax.

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