Substitution

Replace variables with values or expressions to evaluate, simplify, or transform expressions.

📐

Mathematical Definition

Function Evaluation:

f(a) = f(x)|_{x=a}

Substitute

x = a

into function

f(x)

Composition:

f(g(x)) = f(u)|_{u=g(x)}

Substitute

u = g(x)

into function

f(u)

U-Substitution (Integration):

\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du

where

u = g(x)

and

du = g'(x) \, dx

Change of Variables (Multivariable):

\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x}

Code Examples

Single Variable Substitution

Replace variable with number

use mathhook::prelude::*;

let x = symbol!(x);

// Substitute x = 2 into x² + 3x
let expr1 = expr!(x ^ 2 + 3 * x);
let result1 = expr1.substitute(&x, &expr!(2));
// Result: 4 + 6 = 10

// Substitute x = -1 into x³ - 2x + 1
let expr2 = expr!(x ^ 3 - 2 * x + 1);
let result2 = expr2.substitute(&x, &expr!(-1));
// Result: -1 + 2 + 1 = 2

Expression Substitution

Replace with another expression

use mathhook::prelude::*;

let x = symbol!(x);
let y = symbol!(y);

// Substitute x = y + 1 into x² + 3x
let expression = expr!(x ^ 2 + 3 * x);
let substituted = expression.substitute(&x, &expr!(y + 1));
// Result: (y+1)² + 3(y+1)

// Expand for cleaner form
let expanded = substituted.expand();
// Result: y² + 2y + 1 + 3y + 3 = y² + 5y + 4

U-Substitution for Integration

Transform difficult integrals

use mathhook::prelude::*;
use integrals::Integration;

let x = symbol!(x);
let u = symbol!(u);

// Integrate: ∫ 2x·e^(x²) dx
// Let u = x², then du = 2x dx
let integrand = expr!(2 * x * exp(x ^ 2));

// Manual substitution
let u_expr = expr!(x ^ 2);  // u = x²
let integrand_u = integrand.substitute(&expr!(x ^ 2), &u);
// Result: ∫ e^u du = e^u + C

// Back-substitute: e^(x²) + C
let result = expr!(exp(u)).substitute(&u, &u_expr);
// Result: e^(x²) + C

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