# Math Workbench

> Simplify, factor, differentiate, integrate, solve, take limits, series and plots in one box. A small computer algebra system: type commands such as "integrate x*e^x from 0 to 1", "derivative sin(x)^2", "solve x^2 - 5x + 6 = 0", "limit sin(x)/x as x -> 0" or "series cos(x) at 0 order 6". Results are exact (fractions, roots, π) where possible, and the tool says clearly when it falls back to a numeric answer.

Part of usefulweb.site (https://usefulweb.site/). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://mathworkbench.usefulweb.site/ (interactive tool at https://mathworkbench.usefulweb.site/tool)
- JSON API: POST https://mathworkbench.usefulweb.site/api/v1/calculate with a JSON object of inputs (schema: https://mathworkbench.usefulweb.site/api/v1/tool)
- OpenAPI: https://mathworkbench.usefulweb.site/openapi.json
- MCP (Streamable HTTP): https://mathworkbench.usefulweb.site/mcp - tool `math_workbench`

## Inputs
- query (text, required): Command - Commands: simplify, expand, factor, derivative (or d/dx), integrate [from a to b], solve, limit … as x -> a, evaluate … at x = 2, series … at a order n, plot f, g from a to b. Use * or juxtaposition for products (2x), ^ for powers, sqrt(), ln(), log10(), sin(), e, pi.

## Method
- Input is tokenised and parsed with a Pratt parser that understands implicit multiplication (2x, 3(x+1), x sin x), powers of functions (sin^2(x)), |x| and the usual functions; log means the natural logarithm, log10/log2 are explicit.
- All numbers are exact fractions (BigInt rationals); expressions are kept in a canonical form (like terms collected, powers merged, exact values of roots and special angles).
- Derivatives use the sum, product, power and chain rules. Integrals use a table of standard forms, linearity, substitution, integration by parts, partial fractions and the e^(ax)·sin(bx) formula; every antiderivative is verified by differentiating it numerically at several points.
- Definite integrals use F(b) − F(a) when an antiderivative exists and the integrand has no singularity inside the interval, cross-checked by adaptive Simpson (or tanh–sinh quadrature for endpoint singularities and infinite ranges); otherwise the numeric value is shown and labelled as such.
- Limits try direct substitution, then L’Hôpital’s rule on 0/0 and ∞/∞ quotients, degree comparison at infinity, and finally numeric one-sided evaluation with extrapolation (marked as numeric).
- Equations: linear, quadratic, rational, radical and higher polynomial equations (rational root theorem, then Durand–Kerner for the rest); systems of linear equations by exact Gaussian elimination.

## Limits
- A compact computer algebra system, not a full replacement for Mathematica or SymPy: some simplifications (e.g. most trigonometric identities) are not applied, and numeric fallbacks are approximations.