Definite Integral Calculator (Numerical)

Type a function of x and two limits, and this computes ∫ f(x) dx between them numerically, usually to 13 or more significant digits. It uses double-exponential quadrature (tanh-sinh, and its exp-sinh and sinh-sinh forms for infinite limits), which clusters sample points at the ends of the interval so strongly that integrable singularities there cost nothing extra: ∫₀¹ 1/√x, ∫₀¹ ln x and ∫₀^∞ e^(−x²) all come out to full precision. That holds when the singular endpoint is 0: at any other limit, such as 1/√(1−x²) at x = 1, x cannot get closer to the limit than one rounding step of a double, which caps the result near 8 significant digits, and the tool flags it as not fully converged. Substituting u = 1 − x moves the singularity to 0 and restores the lost digits. Limits can be numbers, constants such as pi/2 or sqrt(2), or inf and -inf. The function is written the usual way — powers with ^, implicit multiplication (2x, x sin(x), (x+1)(x−1)), sin, cos, tan and their inverses, sinh, cosh, tanh, exp, ln for the natural logarithm, log for base 10, log2, sqrt, cbrt and abs, with the constants pi and e. The answer comes with an error estimate and the number of function evaluations, and a plot of the integrand over a finite interval shows what was integrated. What it cannot do deserves equal space. The method expects a smooth integrand between the limits: a kink or a jump inside the interval — abs(x) across 0, a step — slows convergence and costs digits, so enter the location as a split point and each piece is integrated separately. A singularity inside the interval, such as 1/x across 0, is reported rather than integrated. Integrals that only converge conditionally over an infinite range, such as sin(x)/x from 0 to infinity, oscillate too slowly for the method and are flagged as not converged; so is a divergent integral like ∫₀¹ 1/x. The result is a number, not a formula: no antiderivative is found.