Integral Calculator

This integral calculator needs no sign-up: enter a function of x and it builds the antiderivative with the rules used and checks it by differentiating. With two limits it also evaluates the definite integral, the area under the graph – exactly, or numerically when no antiderivative is known.

Source: Integration rules: sum, constant-factor and power rules, linear substitution, integration by parts (standard calculus material). Updated: 5 October 2026.

Function f(x)

e.g. x^2 + 3x, sin(2x), e^(3x), 1/(2x+1), x*e^x, x*ln(x), 1/(x^2+1). Allowed: + − * / ^, brackets, sin cos tan exp ln sqrt, e, pi

Lower limit a (optional)

for the definite integral; leave both limits empty if you only need the antiderivative

Upper limit b (optional)

My ToolboxYour inputs are saved in this browser only.

Result

Antiderivative F(x): x·e^x - e^x + C
Check: Derivative of the antiderivative: F′(x) = x·e^x
Definite integral: From 0 to 1: 1 (exact, via the antiderivative)
Steps: Integration by parts (tabular): differentiate the polynomial step by step, integrate the other factor step by step: ∫ x·e^x dx = x·e^x - e^x; The antiderivative holds up to a constant C. It may look different from your textbook answer without being wrong: the check by differentiating shows whether it fits the function.

How it is calculated

An integral reverses differentiation: the antiderivative F of a function f is the function whose derivative is f. The definite integral between two limits a and b measures the area between the graph and the x-axis, with areas below the axis counted as negative.

Which rules the tool applies

Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) for n ≠ −1; for 1/x the result is ln|x|.
Sum and constant factor rules: sums are integrated term by term, numbers in front stay.
Linear inner function: for sin(2x), e^(3x) or (2x+1)³ you divide by the inner derivative.
Integration by parts: products of a polynomial and e^(ax), sin(ax), cos(ax) or ln(x) are solved in tabular form, for example ∫x·eˣ dx = x·eˣ − eˣ.
Substitution: if the integrand contains h(u) together with the derivative u′, u is substituted, for example ∫x·e^(x^2) dx = e^(x^2)/2 or ∫sin(x)·cos(x) dx = sin(x)^2/2.
Fractions: for denominators up to degree 2 the tool uses polynomial division, partial fractions or the arctangent: ∫1/(x^2+1) dx = arctan(x) and ∫1/(x^2−1) dx = ln|x − 1|/2 − ln|x + 1|/2.
More cases: sin², cos² and tan² via identities, eˣ·sin(x) and eˣ·cos(x) by integrating by parts twice, polynomials times roots of ax + b.

Definite integral and check

With limits the tool computes F(b) − F(a). If the rules give no antiderivative, the integral is approximated numerically with adaptive Simpson’s rule and marked as an approximation. Poles inside the interval, such as 1/x from −1 to 1, are detected and reported. The check differentiates the result again; the opposite direction is done by the derivative calculator. The shape of the function is shown in the function plotter.

Graph, error estimate and limits

The graph shows f(x) and shades the area between the limits a and b; parts below the x-axis count as negative. The keys above the graph insert x, ^, √, π and the function names at the cursor. Numerical values come from adaptive Simpson’s rule and are cross-checked with Gauss–Legendre quadrature; the difference is stated in the note. Every antiderivative is verified by differentiating before it is shown. Not supported are, among others, higher powers of sin and cos, denominators of degree 3 or more and roots of quadratic terms – the calculator then reports “no antiderivative found” instead of guessing.

Frequently asked questions

How do I calculate an integral?

First build the antiderivative by integrating each part with the matching rule: the power rule for xⁿ, division by the inner derivative for nested functions, integration by parts for products. For a definite integral then plug in the limits: F(b) − F(a).

What does the “+ C” mean?

A function has infinitely many antiderivatives that differ only by a constant, because the derivative of a constant is zero. That C is therefore added to an indefinite integral and cancels in a definite one.

What is the difference between definite and indefinite integrals?

The indefinite integral is the set of all antiderivatives, a function plus C. The definite integral has limits and gives a number, the signed area between the graph and the x-axis.

Why does the tool sometimes find no antiderivative?

Not every function can be integrated with the standard rules, and some, like e^(x²), have no antiderivative made of elementary functions at all. For a definite integral the tool then calculates numerically.

How do I check the result?

Differentiate the antiderivative: if you get the original function back, the integral is right. The tool shows this check, and for definite integrals it compares the exact value internally with a numerical calculation.

Sources and legal basis

As of: 5 October 2026 · Editorial policy