18.1. Indefinite Integral

Before we'll learn how to calculate definite integral, we'll first learn what is an indefinite integral. Although, as a start, the app can only calculate definite integral but as a way of checking whether or not the definite integral is correct, one needs to find the indefinite integral first.

Since integration is the opposite of differentiation, we'll try to establish a simple proof here on how these two processes become opposites. Consider the function f(x) = 3x^2 - 6x + 4. Then, find the following:

1) The derivative of f(x).
2) The integral of f(x).

To derive f(x), we apply some derivative rules, specifically: power, addition and subtraction rules.
Derivative of f(x): f '(x) = 6x - 6 dx

Note that we found the first and second terms of f ' (x) through the following operations:

First term (6x): Multiply the coefficient 3 and the exponent 2. Then, we subtract 1 from 2.
Second term (-6): Multiply -6 and 1, the exponent of x. Then, subtract 1 from the exponent of x.

First term: Divide 6x by 2.
                        Add 1 to the exponent of the variable x.
Second term: Note that -6 can be written as -6x^0.
                        Divide -6 by 1 and add 1 to 0, exponent of x.
Third term: Since the derivative function has no third
                        term, it means the third term of the original
                        function is a constant.

$\frac{6x^{1+1}}{1+1}=3x^2$

$\frac{-6x^{0+1}}{0+1}=-6x$

$C$

Summarizing the results, we now can write the integral of f(x) into:

$\int \left ( 6x-6 \right )\: dx=3x^2-6x+C$

Observe that the original function, f(x) is in the form of the integral function except that the last term is not a definite value of 4 but a C, which can stands for any constant.

Indefinite Integral

A function g(x) is an indefinite integral, also known as anti-derivative of the function f(x) if:

$g(x)=\int f(x)\, dx$

and

$g'(x)=f(x)$

next: 18.2. definite integral >

list of contents