Integral substitution, also known as u-substitution, is a powerful technique for simplifying integrals. It's essentially the reverse of the chain rule for differentiation. The core idea is to transform a complex integral into a simpler one by introducing a new variable, typically denoted by 'u'.
When to Use It:
This method is most effective when the integrand contains a function and its derivative (or a constant multiple of its derivative). Look for an "inner function" whose derivative is also present in the integrand.
Steps Involved:
Choose the substitution: Identify a part of the integrand that can be represented by a new variable, 'u'. Often, this is the "inner function" of a composite function.
Find the differential: Differentiate your chosen 'u' with respect to 'x' to find du/dx. Then, rearrange this to express dx in terms of du and the original variable (or vice versa, express the original variable terms in terms of 'u' and 'du').
Substitute: Replace all instances of the chosen function with 'u' and dx (or the appropriate differential terms) with their 'u' and 'du' equivalents. The goal is to have an integral solely in terms of 'u' and 'du'.
Integrate: Solve the new, simpler integral with respect to 'u'.
Back-substitute: Replace 'u' with its original expression in terms of 'x' to get the final answer in terms of the original variable.
Example:
Let's find the integral of ∫ x * sin(x²) dx.
Choose u: Let u = x².
Find du: Differentiating u with respect to x gives du/dx = 2x. Rearranging, we get du = 2x dx, or x dx = du / 2.
Substitute: The integral becomes ∫ sin(u) * (du / 2).
Integrate: This simplifies to (1/2) ∫ sin(u) du. The integral of sin(u) is -cos(u). So, the result is (1/2) * (-cos(u)) + C = - (1/2) cos(u) + C.
Back-substitute: Replace u with x² to get the final answer: - (1/2) cos(x²) + C.
This calculator aims to guide you through the substitution process, helping you identify the parts needed for a successful u-substitution. Note that for complex functions, symbolic integration might require advanced algorithms beyond simple pattern recognition.
function calculateIntegral() {
var integrandStr = document.getElementById("integrand").value.trim();
var substitutionVarStr = document.getElementById("substitution_var").value.trim();
var differentialDuStr = document.getElementById("differential_du").value.trim();
var resultElement = document.getElementById("integralResult");
resultElement.textContent = "–"; // Reset previous result
// Basic validation: Ensure fields are not empty
if (integrandStr === "" || substitutionVarStr === "" || differentialDuStr === "") {
alert("Please fill in all fields: Integrand, Substitution Variable (u), and Differential du.");
return;
}
// — Simplified Logic —
// This calculator performs a conceptual substitution based on the inputs.
// It does NOT perform actual symbolic integration.
// It demonstrates how the integrand might look after substitution.
// A true symbolic integration engine is extremely complex.
// The goal is to express the integrand in terms of 'u' and 'du'.
// We assume the user has correctly provided 'u' and 'du'.
// The calculation here is illustrative: showing the structure after substitution.
// Example:
// Integrand: x * sin(x^2)
// Substitution Var (u): x^2
// Differential du: 2*x dx
// To use the calculator conceptually:
// 1. Identify 'u' in your integrand.
// 2. Calculate 'du' and simplify to isolate the remaining parts of the integrand (e.g., 'x dx').
// 3. Input these into the fields.
// For this calculator's output, we'll show a representation of the
// substituted integral structure. We'll aim to replace parts that
// look like 'u' and adjust the differential part.
var substitutedIntegrand = integrandStr;
var adjustedDifferential = differentialDuStr;
// Basic replacement of substitution_var (u)
// This is a very rudimentary string replacement.
// It assumes 'u' is a simple expression and doesn't handle complex cases like u being part of another term.
try {
// Attempt to replace the explicit substitution variable part
// This requires the user to input 'u' and 'du' such that the transformation is clear.
// For instance, if u = x^2, and du = 2x dx, the user enters 'x^2' and '2*x dx'.
// We need to figure out what remains after 'u' and 'du' are accounted for.
// Let's try to infer the structure.
// If integrand is "x*sin(x^2)" and u is "x^2", we want "sin(u)".
// If du is "2*x dx", and we need "x dx", we get "du/2".
// The result would be sin(u) * (du/2).
// More sophisticated parsing would be needed for robustness.
// For this simplified demo, we'll just show a placeholder for the substituted form.
var placeholderResult = "∫ [Original function transformed] du";
// Try a very basic replacement if the substitution variable is found directly.
if (integrandStr.includes(substitutionVarStr)) {
// This is highly speculative and depends on user input format.
// It's hard to programmatically remove 'u' parts without complex parsing.
// Let's simulate the *conceptual* output.
var parts = integrandStr.split(substitutionVarStr);
var transformedTerm = "[Transformed Function of u]"; // Placeholder for the part of the integrand that isn't 'u'
if (parts.length === 2) { // Simple case: [something] + u + [something] or [something] + u * [something]
// Very basic attempt to reconstruct a possible transformed integrand
transformedTerm = parts[0] + "[…]" + parts[1]; // Placeholder
} else if (parts.length === 1 && integrandStr.startsWith(substitutionVarStr)) {
transformedTerm = "[…]" + parts[0];
} else if (parts.length === 1 && integrandStr.endsWith(substitutionVarStr)) {
transformedTerm = parts[0] + "[…]";
}
// Simplistic adjustment for differential – assumes 'du' contains 'dx' and the derivative part.
var differentialPart = differentialDuStr.replace(/dx/i, ").trim();
var adjustmentFactor = "1"; // Default factor
// A very basic attempt to see if the differential part contains the substitution var explicitly
if (differentialPart.includes(substitutionVarStr)) {
// If 'u' is directly in 'du' (unlikely scenario for standard substitution)
adjustmentFactor = "1 / [expression involving u]";
} else {
// More typical: 'du' contains terms related to 'dx' derivative.
// If the original integrand was 'f(g(x)) * g'(x) dx', and u=g(x), du=g'(x)dx
// Then we assume differentialDuStr is like "g'(x) dx".
// We check if the remaining parts of the integrand are covered by the derivative in du.
// This is too complex for simple string manipulation.
// Let's use a placeholder.
adjustmentFactor = "[Factor derived from du]";
}
placeholderResult = `∫ ${transformedTerm.replace(/x/g, 'u')} * ${adjustmentFactor} du`; // Simplistic representation
resultElement.innerHTML = `Conceptual Substituted Integral: ${placeholderResult} + C`;
} else {
// If substitution var isn't directly found, output a generic placeholder.
resultElement.innerHTML = `Conceptual Substituted Integral: ∫ [Transformed Function] du + C`;
}
} catch (error) {
console.error("Error during conceptual substitution:", error);
resultElement.textContent = "Error processing input.";
}
}