# Integration Techniques

In mathematics, integration is the process of finding the area of a region or the volume of a solid object, as a limit of the sums of the areas or the volumes of its enclosed regions. Integral calculus was developed in the 17th century by Isaac Barrow and René Descartes. The most familiar type of integral is the antiderivative, which is a function that, when differentiated, produces the original function. Integrals appear naturally in many situations, and abstract mathematical disciplines often study them. The area of a region or surface is defined to be the limit of the measurement of the lengths of all the line segments that are contained in the region, which in the figure is the shaded region A. The area of a disk is another example of a commonly used integral. In this case, the line segments are the radii of the circular disk. In general, the area of a region or of a surface is the limit of the measurement of the lengths of all the line segments that are contained in the region or that are contained in the surface. The volumes of solids commonly or traditionally thought of as having infinitely many sides are also integrals, but with a volume "V" of a solid the integral is taken over the solid's boundary, and the length of the boundary is not specified. Thus, for example, the volume of a cube is the integral of the area of a unit cube from the faces of the cube to the center of the cube. The area of a region is a non-negative number, and so in this case the result is an approximation of the area of the surface. Integration by substitution is a method used in calculus to approximate the values of integrals. It exploits the fact that for many integrands, the function that generated the integral is of interest, that is, the function that generated the integral is a function of the variable in the integrand. By writing the function in a different form and then integrating, one can approximate the integral of the given function from a nearby point to the point of interest. In this example, the function "f" is the exponential function, which is defined as "f(x)" = "ex", where "x" is the variable. The integral of this function is: The interval of integration is from "x" = 0 to "x" = 1, and the antiderivative of "f" is the natural logarithm. That is, the antiderivative of "f" is "ln"("f"). Since "ln"("f") = 1, by the fundamental theorem of calculus, the integral of "f" from 0 to 1 is: This is an underestimate of the correct value, since it does not take into account the negative values of "f" near 0. However, the difference is small, and the approximation of the integral is good, because the error is proportional to the width of the interval of integration. Integration by parts is a technique used in calculus to find a derivative of a function when both functions are known. The technique is also used to find a second derivative, or integral, of a function, when only the first derivative is known. The technique is based on the fact that the derivative of a product of two functions equals the derivative of the first function multiplied by the derivative of the second function. The derivative of the product of two functions is: where "f" is the function to be evaluated at a particular point and "g" is the function that generates the derivative at the point. The derivative of "f" is the derivative of "g", divided by the width of the interval of integration: The phrase "by parts" refers to the fact that the first and second derivatives must be divided by the width of the interval of integration in order to obtain the derivative of the product of two functions. For example, the derivative of the function "f"("x") = "x" is the derivative of "g"("x") = "x", divided by the width of the interval of integration, or "x" = 1 to "x" = 2. The technique of integration by parts can be used to find the derivative of a function of two variables, for instance "f"("x","y") = "x" sin("y") sin ("x") + "y" sin("x") cos("y"). The derivative of the function "f" is the derivative of the function "g" divided by the width of the interval of integration, or "f"("x","y") = "x" sin("y") sin ("x") + "y" sin("x") cos("y"). The derivative of the function "g" is the derivative of the function "h" divided by