Methods of Integration
Integration is one of the basic operations of calculus. However, its results are not always straightforward; there are some integrals whose solution is not apparent by inspection. Because this is the case, and because closed form and numerical solutions are valuable in many cases, mathematicians have developed several different methods of integration.
In many cases, the integral may be evaluated at all points by direct anti-differentiation. The rules of differentiation are then followed backwards to find a closed form solution of the integral in question. When a definite integral is desired in this case, the anti-differentiated form is simply evaluated at the limits.
In some cases, an apparently difficult integral can be vastly simplified by "u-substitution." In u-substitution, the integral is shifted from the original x variable to a new variable, u, with the limits (if any) changed appropriately and appropriate attention given to the differential of u and how it differs from the differential of x. For example, a common substitution is u = cos x, du = -sin x dx. When the integral has been carried out in simplified form, the original expression is resubstituted for the u value.
Another integration "trick" is integration by parts. Integration by parts is the reverse of the product rule for differentiation, where d/dx (uv) = u (dv/dx) + v (du/dx). Then if the integral is of the form u dv, it's equal to uv minus the integral of v du. If v is a simpler function to integrate than u, this can be a very helpful way of solving the integral. This technique is often used with fractions, where the numerator and denominator are regrouped into equivalent terms.
There are several numerical methods of solving an integral when closed form solutions (as described above) are not feasible but a quantitative answer is desired. The trapezoid or trapezoidal rule is the simplest numerical integration method. The desired area of integration is divided into a selected number of subdivisions. Each of these forms a trapezoid, with the sides of the region being the parallel sides, the axis acting as the base of the trapezoid, and the line connecting the endpoints of the curve on that interval forming the top of the trapezoid. The error in this method may be reduced, as with most numerical integration techniques, by using smaller and smaller subdivisions within the same area. The equivalent result will come of calculating the areas of rectangles whose top was drawn through the midpoint of the function on each interval.
Simpson's rule uses the same type of regional subdivisions as the trapezoid rule, but instead of taking a straight line fit across the function, the top of the interval is approximated to a quadratic equation. The area is then calculated as though the integral was of a parabola instead of over the function in question. Boole's rule uses the same type of quadratic fit but then employs a recursive relationship to make the quadrature more precise. While this is still not an exact calculation of the integral, the error rate can be far reduced.
Rather than selecting values at predetermined, even intervals as the other numerical techniques do, Gauss-Legendre quadrature evaluates the integral based on random points throughout the interval. A table of abscissa and weights allows the random selection to give a fair approximation of the average height of the function throughout the interval (a better and better approximation as more points are selected).
The wide variety among these techniques allows for the person evaluating the integral to select which method is most appropriate to the function and interval and to the application at hand. The advent of computer technology has made the numerical methods more common and more feasible with high precision levels, but each method, exact or approximate, has advantages that keep it in the list of good methods for integration.
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