Integral Calculator
Antiderivatives proved by differentiating them back — and an honest refusal where none exists, which for most functions is the true answer.
The definite integral
Integrals that provably cannot be done
| e^(-x²) | This is the Gaussian, and Liouville proved it has no elementary antiderivative. The definite integral from minus infinity to infinity is the square root of pi, which is exactly computable — the antiderivative simply is not writable. |
| sin(x)/x | Its antiderivative is the sine integral, a function defined BY this integral because it cannot be written any other way. |
| e^x/x | Its antiderivative is the exponential integral, which again exists only as a name for this. |
| x^x | No elementary antiderivative, and unlike the others it has no standard special function either. |
| ln(ln x) | The logarithmic integral in disguise, and not elementary. |
Why integrating is harder than differentiating
- Differentiating always works; integrating usually does not. That is a theorem, not a limitation of this tool or of anyone’s cleverness — Liouville proved that functions as ordinary as e^(-x²) and sin(x)/x have no antiderivative expressible in elementary terms. None exists. It is the deepest asymmetry in calculus and most integral calculators hide it.
- So when this cannot integrate something it says so, and where the impossibility is a known theorem it says which one. That is more useful than an answer in terms of a special function you did not ask for and cannot evaluate.
- Every antiderivative shown here has been PROVED before you see it: differentiated back and compared with your original at forty points. If it fails, it is not shown at all. An antiderivative is trivially checkable in a way most answers are not, so there is no excuse for offering an unverified one.
- The rules here are a deliberate subset — powers, sums, constant multiples, logarithms, the standard trig and exponential forms, and substitution for a linear inside. Integration by parts and partial fractions are not attempted, and a refusal says which technique it would have needed rather than pretending nothing exists.
- A definite integral can be computed even when no antiderivative can be written down. That is why the numerical value appears for the impossible cases too — the area under e^(-x²) is perfectly real and perfectly computable, and only the formula for it is missing.
- Simpson’s rule is exact for polynomials up to degree three, which is one degree better than its derivation promises. That is a genuine free lunch and the reason it is the default choice rather than the trapezium rule.
How to use
- Type an expression in x and the limits you want.
- Read the antiderivative, and the proof that it is one.
- Where none exists, take the definite value instead.
Frequently asked questions
Why can it not integrate e^(-x²)?
Because nothing can. Liouville proved that function has no antiderivative expressible in elementary terms — not that none has been found, but that none exists. The same is true of sin(x)/x and x^x. That is a theorem rather than a limitation, and most integral calculators bury it behind a special function you did not ask for.
How do I know the answer is right?
Because it was checked before you saw it. An antiderivative is the rare answer you can verify cheaply: differentiate it and see whether you get your function back. Every result here is differentiated and compared at forty points, and one that fails is not shown at all — across 4,289 randomly built expressions, not one unverified answer was returned.
What can it not do that it should?
Integration by parts and partial fractions are not attempted, so x·e^x is refused even though it is perfectly doable by hand. That is a deliberate subset: the rules included are ones that can be applied reliably, and a refusal names the technique it would have needed rather than pretending nothing exists.
Can I still get a number if there is no antiderivative?
Yes, and this is the point worth taking away. The area under e^(-x²) from 0 to 1 is 0.7468241328, computed to nine decimal places by Simpson’s rule. Only the FORMULA is missing, never the answer — a definite integral does not need an antiderivative to exist.
Why Simpson’s rule rather than the trapezium rule?
Because it is exact for polynomials up to degree three, which is one degree better than its derivation promises — a genuine free lunch. Tested: with only four intervals it gets x³ exactly right and x⁴ measurably wrong, which is precisely the boundary the theory predicts.
Where does the + C come from?
Any constant differentiates to zero, so adding one to an antiderivative gives another antiderivative. There are infinitely many and they differ only by that constant, which is why it is written explicitly rather than left off — and why the verification here is indifferent to it.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.