Numerical Integration Methods

There is a significant difference between computing a definite integral b f(x) dx and computing an indefinite integral, which may be written in the form a * f(x) dx The result of the first computation is a single number, while the result of the second is a table of numbers. This chapter discusses definite and indefinite integrals. Another distinction should be made. A definite integral may be computed by a single formula, or it may be divided into a number of pieces and the formula applied to each piece. For some practically important, but nevertheless rather special, classes of integrals the solutions can be represented in closed form, i.e., as finite combinations of elementary functions such as polynomials, exponential functions, and logarithms, and of indefinite integrals of such functions. Many other integrals, on the other hand, cannot be solved in this manner. The solution y(x) = о rx t2 dt is not expressible in terms of elementary functions and that is not adequately tabulated. In the face of the obvious limitations of explicit solutions, mathematicians have since the early times of analysis tackled problems in integrations and differential equations by approximate methods of wider applicability.