Definite Integral Calculator

It computes the definite integral of a single-variable function f(x) over a closed interval from a to b, which equals the signed area between the curve and the x-axis on that interval. You enter the function, a lower bound, and an upper bound, and it returns a single numeric value. Areas below the x-axis count as negative, so the result is a net signed area, not the total unsigned area.

It uses adaptive Simpson's rule. Simpson's rule approximates the function with parabolic arcs and the formula (b - a) / 6 times (f(a) + 4 f(m) + f(b)), where m is the midpoint. The interval is split repeatedly and refined only where the function bends the most, so smooth functions converge quickly and tricky regions get extra subdivisions. This gives accurate results without you choosing a step size.

Use x as the variable and combine it with + - * / ^ and parentheses. You can use the constants pi and e and the functions sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, ln, log (base 10), log2, sqrt, cbrt, and abs. For example, enter x^2, sin(x), exp(-x), or 1/(1 + x^2). Multiplication must be explicit, so write 2*x rather than 2x.

No value appears when the bounds are not finite numbers, when the expression cannot be parsed, or when the function is undefined or infinite somewhere on the interval. For instance, 1/x integrated across x = 0 has a singularity at the origin, so the calculator returns nothing rather than a misleading number. Check that your interval avoids points where the function blows up or is undefined.

The calculator follows the standard convention that swapping the limits flips the sign. The integral from b to a equals the negative of the integral from a to b. So entering a lower bound greater than the upper bound returns a negative of the forward result, and the panel notes that the bounds were reversed.

For smooth functions the result is typically accurate to several decimal places, and Simpson's rule is exact for polynomials up to degree three. The calculator also reports an estimated error from its internal convergence check. That estimate is reliable for well-behaved integrands but can understate the true error for functions with very sharp peaks or near-singular behavior on the interval, so treat extreme cases with care.