Basic Concept About the Integral and Its Basic Applications

Calculus is a breakthrough in human’s development and its development is keeping making progresses. This paper mainly talks about the integral, which is a crucial component of calculus. The appearance of integral is much earlier than another part of calculus, the differentiation. The integral helps people to calculate the area, length, and volume of some irregular and sophisticated figures or objects that cannot be simply computed with limited existing tools. The integral appears in many forms, including simpler form and more sophisticated one. The more sophisticated form of integral may comprise two or more compound functions, so that it cannot be calculated simply by the principle of integral. In order to overcome such challenges, mathematicians later developed many specific methods, which are corresponding to some complex forms of integral respectively, such as u-substitution, inverse substitution, integral by parts, improper integral, residue theorem, and Cauchy integral formula. Also, for the examples in this essay, several methods mentioned upon are applied, so that some compound functions including logarithm, exponent, trigonometric function, and infinity are all successfully computed. This essay briefly concludes the history and importance of calculus and the functions of differentiation and integral, detailly introduces the property and operation of integral, provides some examples of integral, and talked about the principle and applications of specific methods of finding integral used in examples is this essay.