| --- |
| license: mit |
| language: |
| - en |
| tags: |
| - mathematics |
| - calculus |
| - symbolic-math |
| - step-by-step |
| - education |
| - java |
| pipeline_tag: other |
| --- |
| |
| # MathSteps: step-by-step calculus in Java |
|
|
| A symbolic maths engine written in **pure Java (no libraries)** that **differentiates** and **integrates** |
| expressions and explains **every step**, like a teacher working on the board. |
|
|
| **Try it:** [MathSteps web app](https://huggingface.co/spaces/unnicr/math-steps). The same Java engine runs |
| in your browser. |
|
|
| ``` |
| > java -jar math-steps.jar int "x e^x" |
| Step 1: Integration by parts: integral u dv = u v - integral v du (choose u by LIATE: logs, inverse trig, algebraic, trig, exponential) |
| u = x, dv = e^x dx, du = 1 dx, v = e^x |
| Step 2: Now integrate v du |
| integral e^x dx |
| - Standard integral: integral e^u du = e^u |
| Standard integral: integral e^u du = e^u => integral e^x dx = e^x |
| Step 3: So u v - integral v du |
| integral x*e^x dx = x*e^x - e^x |
| Step 4: Answer (add the constant of integration C) |
| integral x*e^x dx = x*e^x - e^x + C |
| Step 5: Check: differentiating the answer gives back the integrand |
| d/dx [x*e^x - e^x] = x*e^x (checked) |
| ``` |
|
|
| More worked examples are in [`examples.md`](examples.md). |
|
|
| ## What it can do |
|
|
| **Derivatives:** |
| - **Rules:** constant, power, sum, constant multiple, product, quotient and chain rules. |
| - **Functions:** exponentials (`e^u`, `a^u`), logarithms, trig (`sin cos tan sec csc cot`) and inverse trig (`arcsin arccos arctan`). |
| - **Logarithmic differentiation** for `f(x)^g(x)`, e.g. `x^x`. |
| - **Neat results:** quotient-rule numerators are expanded and simplified. |
|
|
| **Integrals:** |
|
|
| | Technique | Example | |
| |---|---| |
| | Power, exponential, log and trig standard integrals | `x^10`, `2^x`, `1/x`, `sec(x)tan(x)` | |
| | Linear substitution | `sin(3x)`, `e^(5x)`, `(3x+1)^4`, `1/(2x+1)` | |
| | u-substitution (automatic choice of u) | `2x cos(x^2)`, `x e^(x^2)`, `ln(x)/x`, `e^(sqrt(x))/sqrt(x)` | |
| | Substitution inside roots | `x sqrt(x+1)`, `x/sqrt(x+1)` | |
| | Integration by parts (LIATE, repeated) | `x e^x`, `x^3 e^x`, `x^2 ln(x)`, `ln(x)`, `arctan(x)` | |
| | Trig identities | `sin^2(x)`, `sin^3(x)`, `sin^4(x)`, `tan^2(x)`, `sin(x)^3 cos(x)^2` | |
| | Partial fractions (with long division) | `1/(x^2-1)`, `(x+3)/(x^2+3x+2)`, `1/(x(x+1)^2)`, `(x^3+1)/(x-1)` | |
| | Completing the square, giving ln + arctan | `1/(x^2+2x+5)`, `(2x+1)/(x^2+x+1)` | |
| | Inverse-trig forms | `1/sqrt(4-x^2)`, `1/sqrt(x^2+1)` | |
| | e^x · sin / cos | `e^x sin(x)`, `e^(2x) cos(3x)` | |
| | Definite integrals | `sin(x)` from 0 to π = 2 | |
|
|
| **Exact arithmetic:** fractions stay exact (`x^3/3`, not `0.333x^3`), and special values are simplified |
| (`sin(π/6) = 1/2`, `sqrt(8) = 2√2`). |
|
|
| **Honest:** if an integral has no elementary answer (`e^(x^2)`, `sin(x)/x`), it says so instead of guessing. |
|
|
| ## Every answer is checked |
|
|
| - **Derivatives:** compared with a numerical derivative at 10 points. |
| - **Indefinite integrals:** the answer is differentiated back and compared with the original function. |
| - **Definite integrals:** compared with Simpson's rule (4,000 intervals). |
|
|
| The test suite (`test/…/MathStepsTest.java`) runs **227 checks**, and all pass: |
| - 62 derivatives, |
| - 90 integrals, |
| - 10 definite integrals, |
| - 3 integrals that have no elementary answer (correctly rejected), |
| - 62 "print, then read back" round trips. |
|
|
| ## Use it |
|
|
| **Command line** (Java 8 or newer): |
| ``` |
| java -jar math-steps.jar diff "x^2 sin(x)" |
| java -jar math-steps.jar int "1/(x^2+2x+5)" |
| java -jar math-steps.jar defint "x^2" 0 3 |
| java -jar math-steps.jar int "t e^t" --var t |
| java -jar math-steps.jar int "x ln(x)" --json # machine-readable steps (with LaTeX) |
| java -jar math-steps.jar # interactive: d x^3 | i sin(x)^2 | i x^2 from 0 to 1 |
| ``` |
|
|
| **From Java:** |
| ```java |
| import com.unnicr.mathsteps.*; |
| |
| MathSteps.Result r = MathSteps.integral("x^2 ln(x)", "x"); |
| System.out.println(r.answer); // x^3*ln(x)/3 - x^3/9 |
| System.out.println(r.answer.toLatex()); // \frac{x^{3} \cdot \ln\left(x\right)}{3} - \frac{x^{3}}{9} |
| for (Steps.Step s : r.steps.all()) System.out.println(s.title + " :: " + s.text); |
| String json = MathSteps.toJson(r); // for web apps |
| ``` |
|
|
| **Input syntax:** |
| - **Implicit multiplication:** `2x`, `x sin(x)`, `3(x+1)`. |
| - **Powers:** `x^2` or `x**2`. |
| - **Functions:** `sqrt( )`, `e^( )` or `exp( )`, `ln( )` or `log( )`, `sin^2(x)`, `|x|`, `arctan( )` or `atan( )`. |
| - **Constants:** `pi` or `π`. |
| - **Other letters** are treated as constants: `a x^2 + b x + c`. |
|
|
| ## How it works |
|
|
| `Parser` turns text into an expression tree, and `Simplifier` brings it into a canonical form. That means: |
| - exact `Rational` arithmetic, |
| - like terms collected, |
| - powers combined (`x · x^2 = x^3`), |
| - special values applied. |
|
|
| `Differentiator` applies rules recursively and records each one. |
|
|
| `Integrator` tries these strategies in order, rolling back the recorded steps of any strategy that fails: |
| 1. Constant multiple and sum rule. |
| 2. Standard table, with linear substitution. |
| 3. Trig identities. |
| 4. Rational functions: exact polynomial algebra, rational-root factoring, Gaussian elimination for the partial-fraction coefficients. |
| 5. Expansion. |
| 6. u-substitution: tries inner functions as u and accepts u when `integrand / u'` can be written in u alone. |
| 7. Integration by parts. |
|
|
| `Printer` produces both re-parsable plain text and LaTeX. |
|
|
| This is a rule-based symbolic engine, not a neural network. It never "hallucinates" a step, and every answer is verified. |
|
|
| ## Build from source |
|
|
| ``` |
| build.bat (Windows) ./build.sh (Linux/macOS) |
| ``` |
| Both compile for Java 8, run the test suite and build `math-steps.jar`. |
|
|
| ## Limits |
|
|
| - Real-valued, single-variable calculus. |
| - Not yet covered: trig substitution beyond the arcsin/ln forms above, partial fractions with irrational |
| roots (e.g. `1/(x^2-2)`), and repeated irreducible quadratic factors. |
| - When a technique is missing, the solver says it couldn't find a method rather than giving a wrong answer. |
|
|
