Mathematics
Integral calculator: antiderivative and definite integral
Write the function and get its antiderivative, or pick two limits for a definite integral. Under the result you get the method used, and where no elementary antiderivative exists it says so instead of inventing one.
Integration is recognising a derivative backwards
Differentiation is mechanical: there is a rule for every structure and you apply it. Integration is not, which is why it is taught second. There is no procedure that takes any function and returns its antiderivative: there is a repertoire of recognisable forms and a set of techniques for reducing everything else to them. The work is recognition, not execution.
The two techniques that cover nearly every exercise are substitution and integration by parts, and they are the two differentiation rules read backwards. Substitution is the chain rule: a composite function multiplied by the derivative of what is inside it is exactly what the chain rule would have produced. Integration by parts is the product rule rearranged, and it is what you reach for when the two functions have nothing to do with each other.
Then there is the fact that surprises everyone arriving from derivatives: some perfectly ordinary functions have no elementary antiderivative. This is not a gap in the method or in this page — it is a theorem. For e^(−x²), the Gaussian, it is proven that no finite combination of polynomials, exponentials, logarithms and trigonometric functions integrates it. The definite integral exists and is a perfectly respectable number; it is simply obtained another way.
Common mistakes
- Dropping the constant of integration from an indefinite integral. There are infinitely many antiderivatives and the + C is not a formality.
- Applying the power rule with exponent −1. ∫x⁻¹ dx is not x⁰/0: it is ln|x|, and the absolute value matters.
- Changing the variable in a substitution and forgetting the differential. If u = g(x), dx has to become du/g′(x) as well.
- Swapping the limits of a definite integral without changing the sign. Exchanging a and b negates the result.
- Confusing a definite integral with an area. They are the same only where the function is positive.
Frequently asked questions
Why does the answer always carry a + C?
Because two functions differing by a constant have the same derivative: a function has infinitely many antiderivatives and they differ only by that constant. In a definite integral the + C disappears, since it appears twice with opposite signs and cancels.
When do you substitute and when do you integrate by parts?
Substitution is for when you can see a function inside the integrand multiplied by its own derivative: it is the chain rule read backwards. Parts is for a product of two functions of different kinds — typically a polynomial times an exponential, a sine, or a logarithm.
Why does it sometimes not find the antiderivative?
For two different reasons. Sometimes one exists but needs a technique outside this page's repertoire. Sometimes there is none at all: it is proven that for e^(−x²), for sin(x)/x and for others, no finite combination of elementary functions is an antiderivative. The definite integral is still computed, numerically.
What is the difference between the exact and the numeric result?
The exact one comes from the antiderivative evaluated at the limits and is right to the last digit. The numeric one approximates the area by Simpson's rule: accurate enough for ordinary use, but an approximation. The page always says which of the two it gave you.
Can a definite integral come out negative?
Yes, and it is not an error: where the function lies below the x-axis its contribution is negative. If you want the geometric area rather than the integral, split the interval where the function changes sign and add the absolute values.
How this calculation works
Linearity: ∫(a·f + b·g) dx = a∫f dx + b∫g dx. Power: ∫xⁿ dx = xⁿ⁺¹/(n+1) for n ≠ −1, and ∫x⁻¹ dx = ln|x|. Standard: ∫sin x dx = −cos x, ∫cos x dx = sin x, ∫e^x dx = e^x, ∫tan x dx = −ln|cos x|, ∫dx/(1+x²) = arctan x, ∫dx/(A x² + C) = arctan(x√(A/C))/√(AC), ∫ln x dx = x·ln x − x. Linear substitution: ∫f(ax+b) dx = F(ax+b)/a. Substitution: ∫f(g(x))·g′(x) dx = F(g(x)), the chain rule backwards. By parts: ∫u dv = u·v − ∫v du, with u the factor that simplifies on differentiation — the polynomial, or the logarithm. Definite integral: ∫ₐᵇ f dx = F(b) − F(a). Where no elementary antiderivative is found, the definite value is computed by Simpson's rule over 2000 intervals, and the page says so.
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