Indefinite Integral

INTRODUCTION TO THE REVERSE PROCESS OF DIFFERENTIATION

In analysis, the indefinite integral represents the reverse operation of differentiation. It is a process in which we seek a function whose derivative is equal to a given function. Such a function is called an antiderivative or primitive function, and the result of the integration is denoted as the indefinite integral.

DEFINITION OF THE INDEFINITE INTEGRAL

Let 'f' be a function defined on an interval I. The indefinite integral of the function 'f' is denoted by:

∫f(x) dx = F(x) + C,

where:

• F(x) is an antiderivative, for which F′(x) = f(x).
• C is an arbitrary constant, called the constant of integration.

Since the derivative of a constant is zero, every function 'f' has infinitely many antiderivatives, which differ only by a constant.

BASIC RULES OF INTEGRATION

• ∫xⁿ dx = [x^(n+1)] / (n + 1) + C, for n ≠ -1 (Power Rule)
• ∫(1/x) dx = ln|x| + C
• ∫e^x dx = e^x + C
• ∫a^x dx = a^x / ln(a) + C, for a > 0, a ≠ 1
• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C
• ∫(1 / (1 + x²)) dx = arctan(x) + C (or tan⁻¹(x) + C)
• ∫(1 / √(1 – x²)) dx = arcsin(x) + C (or sin⁻¹(x) + C)

Integration respects linearity:
∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx, where 'a' and 'b' are real constants.

EXAMPLE: INTEGRAL OF THE FUNCTION F(X) = 3X² + 2X + 1

We break down the expression and use the basic rules:

∫(3x² + 2x + 1) dx
= 3∫x² dx + 2∫x dx + ∫1 dx
= 3 * (x³/3) + 2 * (x²/2) + x + C
= x³ + x² + x + C.

So, the family of antiderivatives is: F(x) = x³ + x² + x + C.

CONCLUSION

The indefinite integral is the reverse operation of differentiation and leads to a set of functions (a family of functions) that all have the same derivative. By including the constant of integration, we capture all possible solutions. The rules of integration are based on recognizing known forms of functions and their inverse relationship with differentiation laws.