Integral Calculator
Finds the antiderivative of an expression using the power rule, standard forms and linear substitution, and shows the working. Add limits to evaluate a definite integral, or leave them blank for the indefinite form with +C. Every result is checked by differentiating it back before it is shown — if that check fails, the calculator says so instead of guessing.
How it works
How the Integral Calculator works
the part a result on its own leaves out
What an integral measures
The reverse of a derivative: given a rate of change, an integral recovers the original quantity. Graphically, a definite integral is the signed area between a curve and the x-axis — area above the axis counts as positive, area below counts as negative.
Why every antiderivative has a +C
Differentiation destroys constants — the derivative of x² and x² + 5 are identical. Integration cannot recover which constant was there, so it adds a general +C to stand for any of them. A definite integral sidesteps this: the C cancels when you subtract F(a) from F(b).
Why this calculator sometimes declines
Integration is not always reversible the way differentiation is a fixed procedure. Products of two functions of x, or a non-linear expression inside sin, cos or e, need techniques like integration by parts or a substitution the calculator does not attempt. It checks its own answer by differentiating it and comparing to the original — if that does not match, it says so rather than showing a guess.
Common mistakes
Common mistakes with integrals
the errors that actually show up, not every possible slip
Forgetting the +C
x² and x² + 100 have the same derivative, so ∫2x dx could be either — the +C stands for every constant it could have been. Dropping it turns a family of answers into one, which is technically wrong even when the visible part is right.
Dividing by the exponent instead of the new exponent
The power rule raises the exponent by one, then divides by the new exponent, not the old one. ∫x² dx is x³/3 — divide by 3, not by 2.
Treating a non-linear inside like a linear one
∫sin(2x) dx works because 2x is linear — divide by the coefficient 2. ∫sin(x²) dx does not follow the same shortcut, because differentiating cos(x²) would need the chain rule to bring down a 2x, not a constant, so the reverse step is not just 'divide by something'.
Questions
Asked often enough to answer here
answers that change how you work
Does it do definite integrals?
Yes — fill in the lower and upper limit fields and it evaluates F(b) − F(a) using the fundamental theorem of calculus, after finding the antiderivative F.
What if it can't solve my expression?
It says 'beyond what this calculator can do' rather than showing a wrong or unverified answer. That usually means the expression needs integration by parts (a genuine product of two functions of x) or a substitution that is not linear.
How does it check its own answer?
It differentiates the antiderivative it found and compares the result numerically to the original expression at several points. If they do not match, the result is withheld.
What does the +C mean?
It represents any constant, since a constant's derivative is zero and so is invisible to differentiation. Only a definite integral, with two limits, gives one specific number.
Next
Related calculators
what people open straight after this one