Integral Calculator
Solve definite and indefinite integrals online with step-by-step calculus explanations. Supports power rule, standard trigonometric, exponential, and logarithmic functions. Features Simpson's rule numeric integration fallback for non-elementary integrals.
Solve definite and indefinite integrals online with step-by-step explanations.
Use ^ for powers, * for multiplication. Supported: sin, cos, tan, sec, csc, cot, ln, exp, sqrt.
Integrate wrt
Indefinite Integral of f(x)
Antiderivative solved client-side
f(x) = x^2 + 3*x
x^3 / 3 + 3 * x^2 / 2 + C
\int \left[{x}^{2} + 3x\right] dx = \frac{{x}^{3}}{3} + \frac{3 \cdot {x}^{2}}{2} + CWhy Use Our Integral Calculator?
Symbolic Integration Engine
Computes exact algebraic antiderivatives (indefinite integrals) using algebraic rules, power rules, and linear substitutions client-side.
Definite & Indefinite Integration
Switch modes to solve general antiderivatives or evaluate definite integrals over a specific boundary range [a, b].
Simpson’s Rule Numerical Fallback
For complex functions without elementary antiderivatives (e.g. e^(-x²)), the tool automatically computes high-precision numerical integrals.
100% Client-Side Privacy
Every calculation runs entirely in your browser using local React state. No expressions or coordinates are sent to a server.
LaTeX & Copy-Ready Exports
Export formatted integrals as standard LaTeX symbols or plain text expressions for Overleaf, Jupyter, or academic reports with one click.
Automatic Simplification
Expressions are reduced using constant folding, basic algebra, and identity functions to output the cleanest mathematical representation.
Who Uses This Integral Calculator?
Students & Calculus Prep
Verify integration homework problems, check steps for linear substitution, and master the Fundamental Theorem of Calculus.
Physics & Applied Science
Compute physical quantities like total work done (W = ∫ F dx), displacement (s = ∫ v dt), or the center of mass of a distribution.
Probability & Statistics
Calculate the area under a probability density function (PDF) to determine cumulative distributions or expected values.
Financial & Business Modeling
Reconstruct total cost or revenue functions from marginal formulas, and calculate consumer/producer surpluses.
Calculus Educators & Tutors
Generate step-by-step example integration solutions for substitution, power rule, and definite boundaries to present in class.
Numerical Analysis & Research
Verify analytical integral outcomes against Simpson’s rule approximations to validate simulation algorithms.
Understanding Integration, Antiderivatives & Numerical Rules
What Is an Integral?
An integral is a fundamental concept in calculus representing the continuous analog of a sum. Geometrically, a definite integral measures the signed area under a curve y = f(x) between two points a and b. The process of integration is the inverse of differentiation. In physics, integrals are used to accumulate quantities over time or space, such as determining total work from a variable force or finding the mass of a solid with varying density.
Definite vs. Indefinite Integrals
An indefinite integral (or antiderivative) represents a family of functions whose derivative is the integrand. Since the derivative of a constant is zero, we append an arbitrary constant of integration + C to the result: ∫ f(x) dx = F(x) + C. A definite integral computes a specific number by evaluating the antiderivative at two boundaries using the Fundamental Theorem of Calculus: ∫(a to b) f(x) dx = F(b) − F(a).
The Power Rule & Logarithmic Form
The integration power rule is the reverse of differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for any real constant n ≠ −1. For the special case where n = −1, dividing by zero is undefined. Instead, the integral yields the natural logarithm: ∫ (1/x) dx = ln|x| + C. This rule handles standard polynomial integration and functions with fractional or negative exponents.
Integration by Substitution (Linear Forms)
Integration by substitution (u-substitution) is the reverse of the differentiation chain rule. For any composition where the inner function is linear (u = ax + b), its derivative is the constant coefficient a. The differential becomes du = a·dx, which simplifies to dx = du/a. This yields the formula: ∫ f(ax + b) dx = (1/a) · F(ax + b) + C. For example, ∫ sin(3x) dx = −cos(3x)/3 + C.
Numerical Integration (Simpson’s 1/3 Rule)
Many elementary functions (like e^(−x²) or sin(x)/x) do not possess analytical antiderivatives expressible in terms of basic algebraic or transcendental functions. In these cases, we rely on numerical integration.Simpson’s 1/3 Rule approximates the curve with quadratic parabolas over n even intervals, evaluating the function at specific endpoints. The formula is: I ≈ (h/3) · [f(x₀) + 4Σf(x_odd) + 2Σf(x_even) + f(x_n)]. It is highly accurate and commonly used in applied computing.
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Integral Calculator — Frequently Asked Questions
The calculator supports polynomials, trigonometric functions (sin, cos, tan, sec, csc, cot), exponential functions (exp, e^x), natural logarithms (ln, log), square roots, and basic rational/irrational fractions. It handles linear substitutions for functions like sin(3x) or exp(5x-1) and matches inverse trigonometric forms like 1/(x^2 + 4).
Use ^ for exponents (e.g. x^2), * for multiplication (e.g. 3*x or x*sin(x)), and parentheses for grouping. Standard functions are written in lowercase: sin(x), cos(x), tan(x), ln(x), exp(x), sqrt(x). Multiplication is implicitly handled for basic cases like 2x or 3sin(x).
Yes. Switch the mode to "Definite Integral" to input a lower limit (a) and an upper limit (b). The calculator will evaluate the integral over this interval. You can input mathematical constants like pi, e, or expressions like pi/2 or exp(1) directly as bounds.
Many functions (e.g. exp(-x^2) or sin(x)/x) do not have closed-form antiderivatives. For indefinite mode, the calculator will indicate that no symbolic antiderivative was found. For definite mode, it will automatically fall back to high-precision numerical integration using Simpson's 1/3 Rule.
Yes. In the "Variable" input field, you can enter any single letter like t, y, or z. The calculator will treat all other letters in the expression as constants. For example, integrating a*t with respect to t will yield a*t^2/2 + C.
Since the derivative of any constant is zero, there are infinitely many antiderivatives for a function, differing only by a constant. The arbitrary constant of integration "+ C" represents all possible vertical shifts of the antiderivative curve.
The numerical integration utilizes Simpson's 1/3 Rule with 1000 sub-intervals. This approach approximates the curve using quadratic segments, which is highly accurate (yielding double-precision accuracy for most continuous functions).
Yes. The calculator is completely free, does not require sign-up, and runs entirely in your browser. No data or formulas are sent to any remote server, ensuring total privacy for your homework, research, or engineering tasks.