Maths / Calculus
Integral Calculator with Verification and Steps
Find an antiderivative with its constant of integration, or calculate a definite integral between finite bounds. Check the recognised rules and shaded area plot.
Integral Calculator with Verification and Steps: Type a function and get its integral with the method explained, such as partial fractions, trigonometric identities or integration by parts. For a definite integral, give the limits and see the area shaded on a plot. Every answer is checked by differentiating it back and comparing with your function at several points before it is shown. Runs 100% locally in your browser with zero server file uploads.
- Category
- School & study tools
- Runs
- In your browser
- Cost
- Free · no sign-up
- Availability
- Ready to use
Runs entirely in your browser
Real-valued expressions in one variable. Up to 512 characters. Infinite bounds, singular intervals and general piecewise expressions are unsupported.
Antiderivatives and area
An indefinite integral describes a family F(x) + C where F′(x) = f(x). Constants may differ on disconnected domain intervals. A definite integral is signed area; reversing its bounds reverses its sign.
OpenStax Calculus Volume 1, Chapter 5: https://openstax.org/books/calculus-volume-1/pages/5-introduction
Methods and limits
The rule engine performs polynomial long division and partial fractions for clean real factors of denominators of degree at most four. It constructs standard radical primitives, exponential-trigonometric formulas, trigonometric power reductions and repeated integration-by-parts tables. Every result passes derivative-back numerical verification; numerical root reconstruction and sampled verification are not proofs. Library-only working remains an outline.
OpenStax Calculus Volume 2, Chapter 3: https://openstax.org/books/calculus-volume-2/pages/3-introduction
Nerdamer 2.0.0: Apache-2.0; retain the licence and applicable notices. Documentation: https://nerdamer.com/documentation
For related work, use the scientific calculator, graphing calculator and quadratic equation solver.
How to use it
- Enter an integrand, variable and angle unit.
- Choose an indefinite integral, or enter finite bounds for a definite integral.
- Read the verified antiderivative or numerical result, error estimate and working.
Privacy & limitations
Calculations run in your browser. Expressions and bounds are not uploaded.
Related tools
Frequently asked questions
What does no closed form found mean?
Neither the rule engine nor the library returned an antiderivative that passed verification. This does not prove non-existence. Exact recognisers separately identify known non-elementary cases such as exp(-x^2), sin(x)/x, x^x, ln(cos(x)) and 1/ln(x). Finite definite integrals can still be computed numerically.
Are the steps the library’s derivation?
Rule-engine results show the actual division, partial-fraction coefficients, substitutions, identities or parts table used to construct the answer. Library-only results are labelled as an outline; their internal derivation is not claimed.
Can I use infinite bounds or cross a pole?
Infinite bounds and improper integrals are unsupported. A domain check rejects poles and uncertified intervals. The removable hole in sin(x)/x is explicitly filled with its continuous value 1 at zero.
How reliable is the numerical error estimate?
Adaptive Simpson compares successive approximations with an absolute tolerance of 1e-8. Its estimate is not a rigorous bound; highly oscillatory or narrow features can be missed. Calculations stop when the depth or evaluation cap is reached.
Free tool · runs in your browser · no account required