Integral Calculator with Steps
The antiderivative where the form is one that can be recognised, and the definite integral either way.
Results
What this tool does
Integration is differentiation run backwards, and that makes it much harder: there is no rule that always applies, only a collection of forms you learn to recognise. This page works term by term through the standard table — powers, reciprocals, sines and cosines, exponentials, logarithms — and shows which rule it used on each. Every answer comes with its own check: the antiderivative differentiated back, which should return the function you started with. And whether or not the symbolic form is found, the definite integral is always computed numerically as well, so the two can be compared.
Formula
∫ xⁿ dx = x^(n+1) ÷ (n+1) · ∫ₐᵇ f = F(b) − F(a) · Simpson: (h ÷ 3)(f₀ + 4f₁ + 2f₂ + … + fₙ)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
expr | The expression, in x | — |
lower | From | — |
upper | To | — |
PR | The antiderivative | — |
KN | Was the form recognised? | — |
DF | The definite integral | — |
ST | Term · rule used · its antiderivative | — |
NU | The same by Simpson's rule | — |
GP | Gap between the two methods | — |
CK | The answer differentiated back | — |
IN | How the page read it | — |
Worked example
- The expression, in x3x^2 + sin(2x)
- From0
- To1
- The antiderivativex^3 - 0.5cos(2x) + C
- Was the form recognised?Yes
- The definite integral1.7080734183
- Term · rule used · its antiderivative 3x^2 int x^n dx = x^(n+1) / (n+1) x^3 sin(2x) int sin(ax+b) dx = -cos(ax+b) / a -0.5cos(2x)
- The same by Simpson's rule1.7080734183
- Gap between the two methods-1.55431e-15
- The answer differentiated back3x^2 + sin(2x)
- How the page read it3x^2 + sin(2x)
Limitations
- The calculation runs entirely in your browser. The values you type are never sent to a server.
Frequently asked questions
Why does it sometimes say the form is not recognised?
Because integration is genuinely harder than differentiation, and this is not a limitation of the page but of mathematics. There is an algorithm that differentiates any expression; there is no algorithm that integrates any expression, and some perfectly ordinary functions have no elementary antiderivative at all — exp(−x²) is the famous one, and it is why the normal distribution has no closed-form area. This page recognises the forms that are taught: powers, the reciprocal, sines and cosines of a linear argument, exponentials, logarithms. When it meets something outside that list it says so, and gives you the definite integral numerically instead, which always works.
What is the check below the answer?
The antiderivative differentiated back. It should reproduce exactly the function you typed, and if it does not, something is wrong — so you can check the page rather than trusting it. This is the same test you would apply to your own work: an integral is right when differentiating it returns you to where you started. The numerical figure alongside is a second, independent check: the definite integral computed by Simpson's rule, which knows nothing about the antiderivative. When the two agree to ten decimal places, both are almost certainly right.