Integral Function

An integral is the the mathematical value towards which a function goes as the independent variable approaches infinity of a series of sums. In other words, an integral is the consequence you receive if add the area of infinitely small rectangles below the curve. Some integrals are worked out symbolically while others are done numerically.There are some exceptions where it cannot be worked out symbolically. In those cases, one cannot compute them without simply cheating and concluding that the result is another integral.

Integrals of Functions

It is enough to colloquially explain the defined intergral of a function f(x), x ∈ R , as the area specified by the function curve and the horizontal axis in the interval within 2 edges. When the axis is above the curve, the area is considered as negative and positive when it is below the curve. Therefore, the integral defined is represented in notation form as

,it assumes real values

Computation of Integral

An integral can be computed using a limiting procedure i.e., by approximating the curve with horizontal segments and calculating an approximation of the integral as the addition of rectangle areas. If the width of segment reaches 0, the calculated integral converges to the actual value. Closely related to the function derivation there's a symbolic approach. At first, we have to find whether the following properties of linear operators hold good for the integrals

• The sum of the functions of an integral is equal to the sum of the integrals of the individual functions.
• The product of an integral by a constant is equal to the product of the constant by the integral function. Thus, we conclude that the integral operator doesn't give only one number but a whole function. The mental intuition that came to Newton and Leibniz opened the route to a great deal of mathematics and science. As the derivative and the integral are reciprocal operations, they are reversible. This idea rendered in a remarkably simple formula:F(x) = f(x)