An Introduction to Integral Calculus

Introduction to Integral Calculus: 

In Differential Calculus, we have learned how to find the derivative/differential of a differentiable function. The study of the inverse method of differential calculus is the main purpose of Integral Calculus. We now understand this with an example.
Let `f(x)` be a function with `f'(x)` `=d/dx(f(x))` `=3x^2`. From this information, can we determine `f(x)`? The answer is Yes.  Note that `f(x)=x^3` satisfies the condition `d/dx(f(x))` `=3x^2`. We use the theory of Integral Calculus to find `f(x)`.

Definition and Notation:

Let `f(x)` and `F(x)` be two functions of `x` such that 
`d/dx{F(x)}` `=f(x) quad cdots (1)` 
Then `F(x)` is called an Indefinite Integral of `f(x)` with respect to `x`.
The equation (1) can also be written as
`int f(x) dx=F(x)`
Here `int` is the symbol of integration. 


(1). Note that we have
`d/dx{F(x)}` `=f(x)` and `int f(x) dx` `=F(x)`
if and only if
`int d/dx{F(x)} dx =F(x)`
(2). Recall that `d/dx(log x)=1/x`
From the definition of integration, `int 1/x dx` `=log x`
(3). Again we know that `d/dx(cos x)=-sin x`
Hence, we can have `int sin x dx=-cos x`.
(4). We use the theory of integration to find the area of a region bounded by curves.
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