Definite & Indefinite Integration
Last reviewed: April 2026
Integration is the reverse of differentiation — it finds the area under a curve, the accumulation of a quantity, or the antiderivative of a function. The indefinite integral ∫f(x)dx gives a family of functions (plus constant C), while the definite integral ∫ₐᵇf(x)dx gives a specific numerical value representing the area between the curve and the x-axis from a to b. Integration is fundamental to physics (displacement from velocity, work from force), economics (consumer surplus, total cost from marginal cost), probability (cumulative distributions), and engineering. For differentiation, use our Derivative Calculator.
The power rule is the most common: ∫xⁿdx = x^(n+1)/(n+1) + C (for n ≠ −1). Other key rules include ∫eˣdx = eˣ + C, ∫(1/x)dx = ln|x| + C, ∫sin(x)dx = −cos(x) + C, and ∫cos(x)dx = sin(x) + C. This calculator handles polynomial functions using the power rule. For more complex integrals involving substitution, integration by parts, or partial fractions, symbolic math engines like WolframAlpha are recommended.
The fundamental theorem connects differentiation and integration: if F is an antiderivative of f, then ∫ₐᵇf(x)dx = F(b) − F(a). This means we can evaluate definite integrals by finding an antiderivative and computing the difference at the bounds. It's one of the most important results in all of mathematics, linking the concepts of accumulation and rate of change. For equation solving, try our Equation Solver.
| Function | Integral | Rule |
|---|---|---|
| xⁿ | xⁿ⁺¹/(n+1) + C | Power rule (n ≠ -1) |
| 1/x | ln|x| + C | Natural log |
| eˣ | eˣ + C | Exponential |
| sin(x) | -cos(x) + C | Trig |
| cos(x) | sin(x) + C | Trig |
| 1/(1+x²) | arctan(x) + C | Inverse trig |
Integration is one of the two fundamental operations in calculus (alongside differentiation), computing the accumulated total of a quantity that varies continuously. Geometrically, a definite integral represents the signed area between a function's curve and the x-axis over a specified interval. Physically, integration converts rates into totals: integrating velocity over time yields distance traveled, integrating power over time yields energy consumed, and integrating a probability density function over an interval yields the probability of an outcome falling within that range. This calculator evaluates both definite integrals (with specific bounds, producing a numerical result) and indefinite integrals (without bounds, producing a general antiderivative function plus a constant of integration C).
| Function f(x) | Integral ∫f(x)dx | Notes |
|---|---|---|
| xⁿ | xⁿ⁺¹/(n+1) + C | n ≠ −1 |
| 1/x | ln|x| + C | x ≠ 0 |
| eˣ | eˣ + C | Exponential |
| sin(x) | −cos(x) + C | Trig |
| cos(x) | sin(x) + C | Trig |
| 1/(1+x²) | arctan(x) + C | Inverse trig |
| 1/√(1−x²) | arcsin(x) + C | Inverse trig |
When a function does not match a standard formula, systematic techniques transform it into an integrable form. U-substitution — the reverse of the chain rule — replaces a complicated inner function with a single variable: ∫2x·cos(x²)dx becomes ∫cos(u)du = sin(u) + C = sin(x²) + C by setting u = x². Integration by parts — derived from the product rule — handles products of functions: ∫u·dv = uv − ∫v·du. Choosing u and dv follows the LIATE priority: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential — with the first applicable type assigned to u. Partial fraction decomposition breaks rational functions (polynomial divided by polynomial) into simpler fractions, each individually integrable. Trigonometric substitution replaces algebraic expressions with trigonometric identities, simplifying integrals containing √(a²−x²), √(a²+x²), or √(x²−a²). Mastering these four techniques enables evaluation of the vast majority of integrals encountered in science and engineering courses.
The Fundamental Theorem of Calculus connects differentiation and integration: if F(x) is an antiderivative of f(x), then ∫ₐᵇ f(x)dx = F(b) − F(a). This elegant result means definite integral evaluation requires only finding an antiderivative and computing the difference at the bounds — no infinite summation of rectangular areas is needed. The theorem has two parts: the first establishes that the accumulation function (the integral from a to x) is differentiable and its derivative equals the integrand; the second provides the evaluation formula. For example, ∫₀² (3x²+1)dx: the antiderivative is x³+x, evaluated as (8+2)−(0+0) = 10. This result represents the exact area under the curve y = 3x²+1 from x = 0 to x = 2. When analytical antiderivatives cannot be found (many functions have no closed-form integral), numerical methods like Simpson's rule, the trapezoidal rule, or Gaussian quadrature approximate the definite integral to arbitrary precision.
Integration solves problems across every quantitative discipline. In physics, work equals the integral of force over distance (W = ∫F·dx), and electric charge equals the integral of current over time (Q = ∫I·dt). In engineering, the moment of inertia — critical for structural design — is computed by integrating mass times distance squared over the cross-section. In economics, consumer surplus is the integral between the demand curve and the market price. In probability and statistics, the cumulative distribution function is the integral of the probability density function, and expected values are computed by integrating the product of value and probability density. In medicine, the area under the concentration-time curve (AUC) for a drug — computed by integrating plasma concentration over time — determines bioavailability and guides dosing protocols. Understanding when a problem requires integration (finding a total from a rate) versus differentiation (finding a rate from a total) is the essential conceptual skill that makes calculus applicable to real-world quantitative reasoning. See our Statistics Calculator for statistical computations and our Logarithm Calculator for logarithmic function evaluation.
Improper integrals extend definite integration to infinite intervals or functions with infinite discontinuities. Type I improper integrals have infinite bounds: ∫₁^∞ (1/x²)dx = lim(b→∞)[−1/x]₁ᵇ = 0−(−1) = 1. Despite integrating to infinity, the result is finite because 1/x² decreases fast enough. Type II improper integrals involve functions that approach infinity within the integration interval: ∫₀¹ (1/√x)dx = lim(a→0⁺)[2√x]ₐ¹ = 2−0 = 2. Not all improper integrals converge: ∫₁^∞ (1/x)dx diverges to infinity because the harmonic series (its discrete analog) diverges. The comparison test determines convergence by comparing the integrand to a known convergent or divergent function: if 0 ≤ f(x) ≤ g(x) and ∫g converges, then ∫f converges. Improper integrals are essential in probability (the normal distribution integrates over (−∞, ∞)), physics (gravitational and electromagnetic potential energy calculations), and Laplace transforms used in control theory and signal processing.
Many functions encountered in practice — including e^(−x²), sin(x)/x, and most empirically measured data sets — have no closed-form antiderivative. Numerical integration approximates the definite integral using weighted sums of function values at selected points. The trapezoidal rule approximates the area under the curve as a series of trapezoids, with error proportional to the step size squared. Simpson's rule uses parabolic arcs instead of straight lines, achieving fourth-order accuracy — dramatically better precision for smooth functions. Gaussian quadrature selects optimal evaluation points and weights to achieve the highest possible accuracy for a given number of function evaluations. For data tables (experimental measurements at discrete points), the trapezoidal rule is the standard approach because it requires no assumptions about the functional form between data points. Modern computational tools use adaptive quadrature, which automatically refines the step size in regions where the function changes rapidly while using larger steps where the function is smooth — efficiently allocating computational effort for maximum accuracy. Monte Carlo integration offers yet another approach, using random sampling to estimate integrals in high-dimensional spaces where traditional grid-based methods become computationally infeasible — the error decreases as 1/√n regardless of dimension, making Monte Carlo the method of choice for integrals in 10 or more dimensions common in statistical mechanics, financial modeling, and machine learning applications. The choice between analytical and numerical integration depends on whether a closed-form antiderivative exists and whether exact precision is required — for most engineering applications, numerical methods with controlled error bounds provide practically equivalent results with far less mathematical effort than searching for analytical solutions.
Integrals are indispensable tools in engineering and physics, used to calculate quantities that involve continuous accumulation or distribution. In structural engineering, integrals determine the moment of inertia of beams and columns, which governs their resistance to bending and buckling under load. In electrical engineering, integrating current over time yields total charge, while integrating power over time yields total energy consumption. Fluid dynamics relies on surface integrals to calculate flow rates through pipes and channels, and volume integrals to determine the mass of objects with non-uniform density distributions. In probability and statistics, the integral of a probability density function over an interval gives the probability that a random variable falls within that range — the entire field of continuous probability is built on integration.
See also: Limit Calculator · Area Calculator · Mean Median Mode Calculator · Standard Deviation Calculator · Number Sequence Calculator
→ Try u-substitution first. Most integrals that aren't immediately obvious yield to a simple substitution. Look for a function and its derivative appearing together — that's your u.
→ Use definite integrals for area. The definite integral from a to b gives the signed area under the curve. If the function dips below the x-axis, split the integral at the zeros to get total area.
→ Check by differentiating. The derivative of your antiderivative should give back the original function. This is the fastest verification method. Use our Derivative Calculator to verify.
→ Don't forget + C for indefinite integrals. The constant of integration is easily overlooked but represents an entire family of functions. Initial conditions determine the specific value of C.
See also: Derivative Calculator · Graphing Calculator · Area Under Curve · Scientific Calculator
Integration reverses differentiation and computes accumulated quantities from rates of change. The most intuitive application is calculating area under a curve, but this geometric interpretation extends to powerful physical applications: the integral of velocity over time gives displacement, the integral of force over distance gives work, and the integral of power over time gives energy. In probability theory, the integral of a probability density function over an interval gives the probability of an outcome falling within that range. In statistics, cumulative distribution functions are integrals of density functions.
Engineering and science rely on integration for computing volumes of solids of revolution (used in manufacturing to determine material requirements), centroids and moments of inertia (critical for structural engineering and robotics), fluid pressure on surfaces (essential for dam and vessel design), and electromagnetic fields generated by current distributions. Numerical integration methods — trapezoidal rule, Simpson’s rule, and Gaussian quadrature — handle functions that lack closed-form antiderivatives, which is the majority of real-world integrands. Modern computational tools use adaptive numerical methods that concentrate evaluation points where the function changes rapidly, achieving high accuracy with minimal computation. This calculator handles symbolic integration for functions with known antiderivatives and provides step-by-step rule application for educational purposes.