Indefinite Integral
An indefinite integral is the general antiderivative of a function, representing a family of functions that differ only by a constant. Unlike the definite integral, it produces a function rather than a number, and includes an arbitrary constant of integration C that accounts for all possible antiderivatives. Indefinite integrals are the starting point for solving differential equations and computing definite integrals via the Fundamental Theorem of Calculus.
Key Formula
∫ f(x) dx = F(x) + C, where F'(x) = f(x)
LaTeX: \int f(x)\, dx = F(x) + C, \quad \frac{d}{dx}[F(x)] = f(x)
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | Integrand (original function) | dimensionless |
| F(x) | Antiderivative | dimensionless |
| C | Constant of integration (arbitrary) | dimensionless |
Worked Example
Problem
Find the indefinite integral of f(x) = 4x³ − 6x + 5.
Solution
Step 1: Integrate each term using the power rule ∫xⁿ dx = xⁿ⁺¹/(n+1). ∫4x³ dx = 4 · x⁴/4 = x⁴. ∫−6x dx = −6 · x²/2 = −3x². ∫5 dx = 5x. Step 2: Combine and add constant C.
Answer
∫(4x³ − 6x + 5) dx = x⁴ − 3x² + 5x + C
Comparison: Indefinite vs Definite Integral
| Feature | Indefinite Integral | Definite Integral | Example |
|---|---|---|---|
| Output | Function + constant C | Single number | ∫x dx = x²/2+C vs ∫[0,2] x dx = 2 |
| Limits of integration | None | Lower and upper bound a, b | — vs ∫[1 to 3] |
| Constant of integration | Always included (+ C) | Not included | x²/2+C vs 4 |
| Represents | Family of antiderivatives | Net signed area | All curves x²/2+k |
| Used for | Solving ODEs, general form | Area, displacement, probability | Differential equations |
Interactive Tools
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Related Terms
Definite Integral
A definite integral is an integral evaluated over a specific closed interval [a, b], producing a single numerical value that represents the net signed area between the function's curve and the x-axis over that interval. It is defined as the limit of Riemann sums as the number of subintervals approaches infinity. Definite integrals are used extensively in physics for calculating work, displacement, charge, and probability.
Integration
Integration is the process of finding the integral of a function — the reverse operation of differentiation. It computes the accumulation of quantities over an interval and is used to find areas, volumes, total distances, probabilities, and many other quantities in science and engineering. Integration encompasses a wide range of techniques including substitution, integration by parts, partial fractions, and numerical methods.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus establishes the deep connection between differentiation and integration, showing that they are inverse operations. It has two parts: the first part states that if F is the integral function of f, then F is differentiable and F'(x) = f(x); the second part provides a practical method for evaluating definite integrals using antiderivatives. This theorem is arguably the most important result in calculus, unifying two independently developed concepts by Newton and Leibniz.
The word "indefinite" comes from Latin "indefinitus" meaning "not bounded" or "unlimited," reflecting the absence of integration limits. The notation was developed by Leibniz in the late 17th century.