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 $f'(x)=\frac{d}{dx}$(f(x)) = 3x². From this information, can we determine f(x)? The answer is Yes.  Note that f(x)=x³ satisfies the condition $\frac{d}{dx}$(f(x)) = 3x². 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

$\dfrac{d}{dx}{F(x)}$ $=f(x) \quad \cdots (1)$

Then F(x) is called an Indefinite Integral of f(x) with respect to x. The equation (1) can also be written as

∫f(x) dx=F(x)

Here ∫ is the symbol of integration.

Summary:

(1). Note that we have

$\frac{d}{dx}${F(x)} = f(x) and ∫f(x) dx = F(x) if and only if ∫ $\frac{d}{dx}${F(x)} dx = F(x).

(2). Recall that $\dfrac{d}{dx}(\log x)=1/x$

From the definition of integration, ∫1/x dx =log x.

(3). Again we know that $\frac{d}{dx}$(cos x)=-sin x

Hence, we can have ∫sin x dx=-cos x.

(4). We use the theory of integration to find the area of a region bounded by curves.