The different modes of integration all aim at reducing a given integral to one of the fundamental or known integrals. As a matter of fact, there are two principal processes: a. The method of substitution, i. e., a change in the independent variable. b. Integration by parts. In some cases, when the integrand is a rational function it may be broken into partial fractions by the rules of Algebra, and then each part may be integrated by one of the above methods. In some cases of irrational functions, the method of Integration by rationalization is adopted, which is a special case of (a) above. In some cases, integration by the method of Successive Reduction is resorted to, which really falls under case (b). It may be noted that the classes of integrals which are reducible to one or other of the fundamental forms by the above processes are very limited, and the large majority of the expressions, under proper restrictions, can only be integrated by the aid of infinite series.,and in some cases when the integrand involves expressions under a radical sign containing powers of x beyond the second, the investigation of such integrals has necessitated the introduction of higher classes of transcendental functions such as elliptic functions, etc. In fact, integration is, on the whole, a more difficult operation than differentiation. The Differential Calculus gives general rules for differentiation, but Integral Calculus gives no such corresponding general rules for performing the inverse operation. Integration is essentially a tentative process. In fact, so simple an integral in appearance as int (x)^(1/2) cos x dx, or int (1/x)(sin x) dx can not be worked out; that is there is no elementary function whose derivative is (x)^(1/2) cos x, or (1/x)(sin x), though the integrals exist. There is quite a large number of integrals of this types. [Extracted from INTEGRAL CALCULUS, Das, B. C., B. N. Mukherjee; article 1.6]
Posted on: Mon, 03 Feb 2014 15:52:22 +0000
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