unnicr/math-steps

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. 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.

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:

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.