How this integral calculator works
This is a definite integral calculator: it returns one number, the signed area between your curve and the x-axis from a to b. It never finds an antiderivative — instead it evaluates f(x) at 1,001 evenly spaced points and combines them with composite Simpson’s rule, which is why it can integrate functions that have no closed-form antiderivative at all (try e^(-x^2) from 0 to 1). The honest trade-off: if you need the indefinite integral — the symbolic antiderivative, with steps — that takes a computer algebra system, not a numerical method. No calculator that works this way can show algebra steps, because there aren’t any.
Supported syntax: numbers, x, the operators + - * / ^, parentheses, the functions sin cos tan asin acos atan sqrt abs ln log exp (log is base 10, ln is natural), and the constants pi and e. Multiplication must be explicit — 2*x, not 2x.
The formula
Here n is the number of subintervals (we use 1,000), h is the width of each one, and x0 … xn are the sample points from a to b. The friendly version: Simpson’s rule slices your interval into narrow strips, and instead of approximating the curve in each strip with a flat top (rectangles) or a slanted top (trapezoids), it fits a small parabola through each set of three neighboring points. Parabolas hug curves so well that the error shrinks with the fourth power of the strip width — halve the strips and the error drops roughly 16-fold. That’s why 1,000 strips is comfortable overkill for anything smooth.
Worked example
∫01 x² dx — the antiderivative is x³/3, so the exact answer is 1/3. The calculator returns 0.333333, matching to every displayed digit (Simpson’s rule is actually exact for polynomials up to degree 3).
∫0π sin(x) dx — the antiderivative is −cos(x), so the exact answer is −cos(π) − (−cos(0)) = 1 + 1 = 2. Enter sin(x) with bounds 0 and pi and you’ll get 2.000000.
Common antiderivatives worth memorizing
For hand integration (and for checking this calculator against exact answers), these five cover a remarkable share of homework:
| f(x) | ∫ f(x) dx | Watch out for |
|---|---|---|
| xn (n ≠ −1) | xn+1⁄(n+1) + C | Fails at n = −1 — that case is the next row |
| 1⁄x | ln|x| + C | The absolute value matters for x < 0 |
| ex | ex + C | Its own antiderivative — enjoy it |
| sin(x) | −cos(x) + C | The minus sign, every single time |
| cos(x) | sin(x) + C | No minus sign here — sin/cos are asymmetric |
When numerical integration lies (and how we avoid it)
Numerical methods fail loudly in one case and quietly in another. The loud case is a singularity: integrate 1/x from −1 to 1 and a sample point lands on the vertical asymptote at 0. Some calculators average their way to a plausible-looking garbage number; this one checks every sample for infinities and domain errors (sqrt of a negative, asin beyond ±1) and tells you the integral is improper instead. The quiet case is undersampling: a function oscillating thousands of times between your bounds can wiggle right between the sample points. With 1,000 subintervals you’re safe up to a few hundred oscillations; past that, split the interval and integrate the pieces.