Integral of a Function

Integral of a Function

The result of either a definite integral or an indefinite integral.

Examples: indefinite integral of x² = (1/3)x³ + C; definite integral from 2 to 5 of x² = 117/3 = 39

Key Formula

\int_a^b f(x)\,dx \quad \text{(definite)} \qquad \int f(x)\,dx = F(x) + C \quad \text{(indefinite)}

Where:

$f(x)$ = The function being integrated (called the integrand)
$x$ = The variable of integration
$a, b$ = The lower and upper limits of integration (definite integral only)
$F(x)$ = An antiderivative of f(x), meaning F'(x) = f(x)
$C$ = The constant of integration, representing an arbitrary constant (indefinite integral only)
$dx$ = Indicates integration is performed with respect to x

Worked Example

Problem: Find the definite integral of f(x) = 3x² from x = 1 to x = 4.

Step 1: Write the integral with the given limits.

\int_1^4 3x^2\,dx

Step 2: Find the antiderivative using the power rule: increase the exponent by 1 and divide by the new exponent.

\int 3x^2\,dx = 3 \cdot \frac{x^3}{3} = x^3

Step 3: Apply the Fundamental Theorem of Calculus: evaluate the antiderivative at the upper limit and subtract its value at the lower limit.

\Big[x^3\Big]_1^4 = 4^3 - 1^3

Step 4: Compute the arithmetic.

64 - 1 = 63

Answer: The definite integral equals 63. This represents the exact area under the curve y = 3x² between x = 1 and x = 4.

Another Example

This example shows an indefinite integral (no limits), uses the sum rule to handle two terms of different types (polynomial and trigonometric), and includes the constant of integration C—which the definite integral example did not require.

Problem: Find the indefinite integral of f(x) = 6x + cos(x).

Step 1: Write the indefinite integral.

\int (6x + \cos x)\,dx

Step 2: Split the integral using the sum rule: the integral of a sum equals the sum of the integrals.

\int 6x\,dx + \int \cos x\,dx

Step 3: Integrate each term separately. For 6x, use the power rule. For cos(x), recall that the antiderivative of cos(x) is sin(x).

6 \cdot \frac{x^2}{2} + \sin x = 3x^2 + \sin x

Step 4: Add the constant of integration C.

3x^2 + \sin x + C

Answer: The indefinite integral is 3x² + sin(x) + C.

Frequently Asked Questions

What is the difference between a definite integral and an indefinite integral?

A definite integral has upper and lower limits and produces a single number, often representing area, displacement, or accumulated quantity. An indefinite integral has no limits and produces a family of functions (antiderivatives) that differ by a constant C. The Fundamental Theorem of Calculus connects them: you evaluate an indefinite integral at two points to compute a definite integral.

Why do you add '+ C' to an indefinite integral?

The constant of integration C accounts for the fact that many different functions share the same derivative. For example, x², x² + 5, and x² − 3 all have derivative 2x. When you reverse differentiation, you cannot know which constant was originally present, so you write + C to represent all possibilities.

What does the integral of a function actually represent?

Geometrically, a definite integral represents the signed area between the graph of the function and the x-axis over a given interval. 'Signed' means areas below the x-axis count as negative. In applications, integrals represent accumulated quantities such as total distance, total mass, work done by a force, or probability over a range.

Definite Integral vs. Indefinite Integral

	Definite Integral	Indefinite Integral
Notation	$\int_a^b f(x)\,dx$	$\int f(x)\,dx$
Result	A number	A function (plus C)
Limits of integration	Has lower limit a and upper limit b	No limits
Constant of integration	Not needed (cancels during evaluation)	Required (+ C)
Interpretation	Signed area under the curve, accumulated quantity	General antiderivative of the function
Example	$\int_0^2 x\,dx = 2$	$\int x\,dx = \frac{x^2}{2} + C$

Why It Matters

Integration is one of the two central operations of calculus (alongside differentiation) and appears throughout advanced math, physics, engineering, and economics. You will encounter integrals whenever you need to find areas, volumes, average values, total accumulated change, or probabilities from a density function. Mastering the integral of a function is essential for AP Calculus, university-level STEM courses, and any field that models continuous change.

Common Mistakes

Mistake: Forgetting the constant of integration C on indefinite integrals.

Correction: Every indefinite integral represents a family of functions that differ by a constant. Always write + C unless you are evaluating a definite integral or solving for a specific antiderivative using an initial condition.

Mistake: Applying the power rule incorrectly by forgetting to divide by the new exponent.

Correction: The power rule for integration states that ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1). Students often raise the exponent but forget to divide. For example, ∫x³ dx = x⁴/4 + C, not x⁴ + C.

Related Terms

Definite Integral — Integral with limits yielding a numerical value
Indefinite Integral — Integral without limits yielding an antiderivative
Integration Methods — Techniques such as substitution and integration by parts
Integral Rules — Standard rules like power, sum, and constant rules
Integral Table — Reference list of common antiderivatives
Antiderivative — A function whose derivative is the integrand
Fundamental Theorem of Calculus — Links definite integrals to antiderivatives
Derivative — The inverse operation of integration