Question

Question #89633

Answer 1

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.