Question #89633

1 Answer
Sep 11, 2016

Integration refers to the inverse process of differentiation. Allow me to illustrate it briefly in the section below.

Explanation:

So example, we have a single valued and differentiable function #f(x)#

The differential coefficient or simply the derivative of #f(x)# is defined as,

#f_1(x) = (df)/dx# which can be defined from the first principle. Where the subscript 1 denotes that it is the first derivative of #f#.

Now, let #F(x) = f_1(x)# denote the derivative of #f(x)#

We define the integral of #F(x)# as,

#int F(x)dx = f(x) + C# where #C# is called the constant of integration.

#C# is completely arbitrary and may be defined in some cases by the boundary conditions of the particular problem.

Since, the integral of #F(x)# can have an indefinite number of forms just differing by the value of the constant #C#, this is called the indefinite integral of #F(x)#.

So far we are done with the basic definition of integration.

I would like to extend the discussion to the definition of the definite integral.

The definite integral of #F(x)# from #a# to #b# is simple defined as,

#int_a^bF(x)dx = [f(x)]_a^b#

Now, the fundamental theorem of integral calculus states that,

#int_a^bF(x)dx =f(b) - f(a)#.

There are simple rules for determining the integral of a function. We you are dealing with the indefinite integral, just put a constant and if you're dealing with the definite integral, use the fundamental theorem of integral calculus.

A more rigorous and geometric interpretation of definite integral is provided in most elementary calculus textbooks. Please consult one if you like.