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Integral — Definition, Formula & Examples

Integral

As a noun, it means the integral of a function.

As an adjective, it means "in the form of an integer." For example, saying a polynomial has integral coefficients means the coefficients of the polynomial are all integers.

 

 

See also

Integration methods, integral rules, integral table

Key Formula

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\, dx = F(b) - F(a)
Where:
  • f(x)f(x) = The function being integrated (the integrand)
  • aa = The lower limit of integration
  • bb = The upper limit of integration
  • F(x)F(x) = An antiderivative of f(x), meaning F'(x) = f(x)
  • dxdx = Indicates integration is 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.
∫143x2 dx\int_1^4 3x^2\, dx
Step 2: Find the antiderivative F(x). The antiderivative of 3x² is x³, since the derivative of x³ is 3x².
F(x)=x3F(x) = x^3
Step 3: Apply the Fundamental Theorem of Calculus: evaluate F(b) − F(a).
F(4)−F(1)=43−13=64−1=63F(4) - F(1) = 4^3 - 1^3 = 64 - 1 = 63
Answer: The definite integral equals 63. This represents the exact area under the curve f(x) = 3x² between x = 1 and x = 4.

Another Example

Problem: Find the indefinite integral of f(x) = 6x + 5.
Step 1: Write the indefinite integral.
∫(6x+5) dx\int (6x + 5)\, dx
Step 2: Integrate each term separately using the power rule: increase the exponent by 1, then divide by the new exponent.
∫6x dx+∫5 dx=3x2+5x\int 6x\, dx + \int 5\, dx = 3x^2 + 5x
Step 3: Add the constant of integration C, since any constant vanishes when you differentiate.
3x2+5x+C3x^2 + 5x + C
Answer: The indefinite integral is 3x² + 5x + C.

Frequently Asked Questions

What is the difference between a definite and an indefinite integral?
A definite integral has upper and lower limits of integration (like a and b) and produces a specific number, often representing the area under a curve on that interval. An indefinite integral has no limits and produces a family of functions (antiderivatives), written with a '+ C' constant of integration.
What does an integral actually measure?
An integral measures the total accumulation of a quantity. Geometrically, the definite integral of a positive function gives the area between the curve and the x-axis. More broadly, integrals can represent total distance, total mass, volume, work, or any quantity that accumulates continuously.

Integral (Antiderivative) vs. Derivative

The derivative measures the instantaneous rate of change of a function, while the integral reverses this process—it accumulates the quantity whose rate of change is given. The Fundamental Theorem of Calculus formally links the two: differentiation and integration are inverse operations. If you differentiate the integral of a function, you get the original function back.

Why It Matters

Integrals are essential throughout science, engineering, and economics. Physicists use integrals to compute work done by a force, electrical engineers use them to analyze signals, and economists use them to find consumer surplus from demand curves. Whenever you need to add up infinitely many infinitesimally small contributions—areas, volumes, probabilities—you need an integral.

Common Mistakes

Mistake: Forgetting the constant of integration (+ C) on indefinite integrals.
Correction: An indefinite integral represents a family of functions that all share the same derivative. The constant C accounts for the fact that many different functions have the same derivative (e.g., x² + 3 and x² − 7 both have derivative 2x). Always include + C.
Mistake: Confusing the integral with the antiderivative evaluated at a single point.
Correction: A definite integral requires you to evaluate the antiderivative at both limits and subtract: F(b) − F(a). Evaluating at only one endpoint gives a wrong answer.

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