General Definition of Integral
Smaridasa Mathematics – Hello everyone, how are you? Hopefully you are doing well. On this occasion, we will discuss something called integral. What is an integral? And how do we calculate it? We will learn it step by step, starting with the General Definition of Integral before moving on to more in-depth topics. However, it is recommended that you first master the topic of function derivatives, because integrals are closely related to derivatives.
Relationship Between Derivative and Integral
- \( F(x) \) → derivative → \( f(x) \)
- \( F'(x) \) → integral
Suppose \( f \) is the derivative function of a continuous function \( F \) on a certain domain. For every \( x \) in that domain, \( F'(x) = \frac{dF(x)}{dx} = f(x) \) holds, meaning the derivative of \( F(x) \) is \( f(x) \).
Observe the following function \( F(x) \) and its derivative \( f(x) \):
\[ F(x) = x^2 \quad \rightarrow \quad F'(x) = f(x) = 2x \]
\[ F(x) = x^2 + 3 \quad \rightarrow \quad F'(x) = f(x) = 2x \]
\[ F(x) = x^2 - 5 \quad \rightarrow \quad F'(x) = f(x) = 2x \]
\[ F(x) = x^2 + \sqrt{3} \quad \rightarrow \quad F'(x) = f(x) = 2x \]
\[ F(x) = x^2 + c \quad \rightarrow \quad F'(x) = f(x) = 2x \quad (c \text{ is a constant}). \]
From this fact, a question arises: How can we determine the function \( F \) such that for every \( x \) in the domain of \( F \), \( F'(x) = f(x) \) holds? An operation used to determine \( F \) is the inverse of the derivative (differential) operation. The inverse of the derivative operation is called integral. Integral is also known as antiderivative or antiderivative. In the example above, if \( F(x) \) is the integral of \( f(x) = 2x \), then \( F(x) = x^2 + c \), where \( c \) is a real constant.
Definition of Integral
If \( F(x) \) is a general function such that \( F'(x) = f(x) \), then \( F(x) \) is the antiderivative or integral of \( f(x) \).
The integration of function \( f(x) \) with respect to \( x \) is denoted as follows:
\[ \int f(x) \, dx = F(x) + c \]
Explanation:
\( \int \) = integral notation (introduced by Leibniz, a German mathematician)
\( f(x) \) = integrand function (the function whose antiderivative/integral is sought)
\( F'(x) \) = general integral function such that \( F'(x) = f(x) \)
\( c \) = constant of integration.
Example Problems:
1) Determine the result of the following integrals:
a) \( \int 2x \, dx \)
b) \( \int (x + 3) \, dx \)
Solution (a):
\[ \int 2x \, dx = x^2 + c \]
Because the derivative of \( x^2 + c \) is \( 2x \).
Solution (b):
\[ \int (x + 3) \, dx = \frac{1}{2}x^2 + 3x + c \]
Because the derivative of \( \frac{1}{2}x^2 + 3x + c \) is \( x + 3 \).
Note: In further integral calculations, we will use existing formulas, meaning we do not need to derive them again as in the examples above.
Subtopics in Integral
The subtopics we will study in integral are:
- Definition of integral
- Difference between definite and indefinite integrals
- Indefinite integral of algebraic functions
- Determining curve equations using integrals
- Indefinite integral of trigonometric functions
- Integration techniques:
- Algebraic substitution
- Integration by parts
- Trigonometric substitution
- Integration by partial fractions
- Definite integrals:
- Riemann sums
- Fundamental theorem of calculus
- Properties of definite integrals
- Applications of integrals:
- Finding area under a curve
- Determining arc length of a curve
- Finding volume of solids of revolution
The integration of function \( f(x) \) with respect to \( x \):
\[ \int f(x) \, dx = F(x) + c \]