Mathematics

INTEGRAL CALCULUS

A. Basic Concepts:

Integration has 2 aspects:

(1) as the inverse of differentiation

(2) as the limit of summation

The corresponding results are:

(1) the indefinite integral or anti-derivative: the inverse of differentiation process consists of function y = f(x) which is regarded as the derivative f ‘(x) for a primitive y = f(x) dx = f(x) + C where C = constant of integration.

(2) the indefinite integral of the summation: the summation process starts with formally identical infinitesimal (very small) element which, in efforts are increment ∆y = f(x) ∆x for the continuous function y = f(x).

B. Anti-derivation:

Anti-derivation is the direct process of derivatives in its inverse. Anti-derivation

(1) has no formal method

(2) does not always have a formal result. The only way to

obtain the indefinite integral of y = ∫ f(x) dx is to recognize the given integrand f(x) as the derivative of the results y = f(x) + C. The so-called methods of anti-derivation are clearly methods of putting the integrand in a recognized form. Similarly, the so-called integration formulas are actually derivation formulas given in reverse. To make an integral recognizable as a derivative, the following operations may be tried:

(a) dividing

(b) expanding

(c) factoring

(d) multiplying by one in the form of an integrating

factor or a rationalizing conjugate

(e) adding zero to complete the square or to

decompose

(f) substituting

(g) integrating by parts

(h) resolving into partial fractions

C. Basic Integration Formulas:

1. ( du = u + C 2. ( a du = a ( du = a u + C

3, ( un du = un+1 + C 4. ( du = ln u + C

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