Integral Calculator — Indefinite, Definite, Double, Triple & Numerical
Type any function and get an instant, fully worked integral — indefinite or definite, single, double, or triple — with every rule applied and explained. Built for calc 1 & calc 2 students, engineers, and anyone who needs to see the working, not just the answer.
Step-by-Step Solution
Whether you're checking a Calc 2 homework answer, evaluating a definite integral for a physics problem, or trying to remember exactly how integration by parts applies to x·eˣ, the AceCalculator integral calculator gives you an answer you can trust — and, more importantly, shows you exactly how it got there.
Every result is built from the same rules you'd apply by hand: the power rule for integrals, u-substitution, integration by parts, and the Fundamental Theorem of Calculus. AceCalculator applies them in order, labels each one, and lets you see the calculation unfold instead of just handing you a final expression.
This tool covers five distinct modes in one place — indefinite integrals, definite integrals, double integrals, triple integrals, and numerical integration via Simpson's and trapezoidal rules — so you don't need five different calculators bookmarked. Everything runs directly in your browser. No downloads, no accounts, no data sent anywhere.
What You Can Do With This Calculator
Indefinite Integrals — Antiderivatives With +C
The indefinite integral calculator mode handles single-variable functions of any complexity: polynomials, roots, trig functions, exponentials, and logarithms, combined freely, always returning the constant of integration.
- Power rule and reverse chain rule for polynomial and nested terms
- Standard exponential, logarithmic, and trig integral forms
- u-substitution for functions with a linear inner term
- Every result includes "+ C"
Definite Integral Calculator With Steps
Need a number, not just an expression? The definite integral calculator finds the antiderivative, then applies the Fundamental Theorem of Calculus to evaluate it between your two bounds — including improper integrals with an infinite or undefined endpoint.
- Evaluates F(b) − F(a) automatically
- Accepts numeric or symbolic bounds (π, e, fractions)
- Flags and approximates improper integrals
Double Integral Calculator
For functions of two variables, the double integral calculator integrates f(x, y) over a region — a rectangle, or a region where the y-bounds are themselves functions of x — using nested numerical integration.
- ∬ f(x,y) dy dx over rectangular or variable-bound regions
- y-bounds can be entered as constants or expressions in x
- Useful for area, volume, and mass/centroid problems
Triple Integral Solver
The triple integral calculator extends the same approach to three variables, evaluating f(x, y, z) over a solid region defined by constant or variable bounds.
- ∭ f(x,y,z) dz dy dx via nested numerical integration
- z-bounds can be entered as constants or expressions in x and y
- Useful for volume and 3D mass/moment problems
Numerical Integration — Simpson's & Trapezoidal Rule
When a function has no closed-form antiderivative, switch to Numerical mode to approximate the definite integral using Simpson's Rule or the Trapezoidal Rule with a chosen number of subintervals.
- Simpson's Rule for smooth, quickly-converging estimates
- Trapezoidal Rule for a simpler, linear approximation
- Adjustable subinterval count (n) to control precision
Full Step-by-Step Breakdown
Every mode returns a numbered list of steps, each one tagged with the specific rule used — Power Rule, u-Substitution, Integration by Parts, or the Fundamental Theorem of Calculus — plus the formula being applied.
- Rule-by-rule breakdown, not just a final answer
- Formulas shown alongside each applied step
- Designed to match how the topic is taught in class
The Rules Behind the Calculator
Every step this calculator produces comes from one of the standard integration rules below. Understanding these is the fastest way to check the working by hand.
| Rule | Formula | Used For |
|---|---|---|
| Power Rule for Integrals | ∫xⁿ dx = x⁽ⁿ⁺¹⁾/(n+1) + C | Polynomial terms |
| Sum / Difference Rule | ∫[f ± g] dx = ∫f dx ± ∫g dx | Multi-term expressions |
| u-Substitution | ∫f(g(x))g′(x) dx = ∫f(u) du | Nested / composite functions |
| Integration by Parts | ∫u dv = uv − ∫v du | Products of two functions |
| Fundamental Theorem of Calculus | ∫ₐᵇ f(x) dx = F(b) − F(a) | Definite integrals |
| Simpson's Rule | ∫ₐᵇ f(x) dx ≈ (h/3)[f(x₀)+4∑f(odd)+2∑f(even)+f(xₙ)] | Numerical integration |
Why AceCalculator Is Different And Better
Most integral calculators online give you an answer and nothing else. AceCalculator focuses on showing the working, matching classroom notation, and staying fast on any device.
- ✓ Full rule-by-rule step-by-step solutions, every mode
- ✓ Five calculators in one: indefinite, definite, double, triple, numerical
- ✓ Clean, classroom-style notation
- ✓ Mobile-friendly and responsive on any device
- ✓ No downloads, no accounts, no signup required
- ✓ Runs entirely in your browser — nothing is stored or sent to a server
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Frequently Asked Questions
Yes. Every mode — indefinite, definite, double, triple, and numerical integration — is free to use with no account, signup, or download required.
Yes. Each result comes with a numbered list of steps, and each step is labelled with the specific rule applied (Power Rule, u-Substitution, Integration by Parts, or the Fundamental Theorem of Calculus) along with the formula used.
An indefinite integral returns a general antiderivative plus a constant of integration, C. A definite integral evaluates that antiderivative between two bounds using the Fundamental Theorem of Calculus, returning a single numeric value.
Yes. Switch to Double or Triple mode, enter a multivariable function, and set the bounds for each variable — bounds can be constants or expressions in the outer variables. The calculator uses nested numerical integration to evaluate the result.
Yes. In Definite mode, if a bound is infinite or the function is undefined at an endpoint, the calculator approximates the limiting value numerically and labels the result as an improper-integral approximation.
Numerical mode uses Simpson's Rule or the Trapezoidal Rule to approximate a definite integral directly from function values — useful when a closed-form antiderivative is difficult or impossible to find, unlike tools that only return a symbolic result.
The calculator applies the same integration rules taught in standard calculus courses, and numerical modes converge closer to the true value as the subinterval count (n) increases. As with any automated tool, always double-check results for high-stakes coursework or professional work.
Important Disclaimer
AceCalculator's integral calculator is provided for informational and educational purposes only. While every step follows standard integration rules, results — especially numerical approximations — are not a substitute for verification by a qualified instructor or professional where accuracy is critical (exams, published research, or engineering work). See our full Disclaimer and Terms and Conditions.