Indefinite Integral Calculator
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Indefinite Integral Calculator
Your integral will appear here.
Understanding Indefinite Integrals
An indefinite integral, also known as an antiderivative, is a function whose derivative is the original function. In simpler terms, it's the reverse process of differentiation. When we differentiate a function, we find its rate of change. When we integrate a function indefinitely, we find a family of functions that all have the same rate of change.
The indefinite integral of a function f(x) is denoted as ∫ f(x) dx. The result is a function F(x) + C, where F'(x) = f(x) and C is the constant of integration. The constant C is crucial because the derivative of any constant is zero. Therefore, there isn't a single unique antiderivative, but rather an infinite family of functions differing only by a constant.
Key Concepts and Formulas:
- Power Rule: For any real number
n ≠ -1, the integral of x^n is (x^(n+1))/(n+1) + C.
- Constant Multiple Rule:
∫ c * f(x) dx = c * ∫ f(x) dx, where c is a constant.
- Sum/Difference Rule:
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.
- Common Integrals:
∫ x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)∫ (1/x) dx = ln|x| + C∫ e^x dx = e^x + C∫ a^x dx = (a^x)/ln(a) + C∫ sin(x) dx = -cos(x) + C∫ cos(x) dx = sin(x) + C∫ sec^2(x) dx = tan(x) + C∫ c dx = cx + C
How This Calculator Works (Simplified):
This calculator uses a simplified approach to handle basic polynomial and common transcendental functions. It attempts to parse common function formats and apply standard integration rules. Due to the complexity and infinite variety of mathematical functions, this calculator is designed for illustrative purposes and may not handle all possible inputs or advanced integration techniques (like integration by parts, trigonometric substitution, or complex series).
For complex functions, numerical integration methods or advanced symbolic computation software are typically required.
Use Cases:
- Physics: Calculating position from velocity, velocity from acceleration.
- Engineering: Determining total quantity from a rate of change.
- Economics: Calculating total cost or revenue from marginal cost or revenue functions.
- Mathematics Education: Helping students understand and verify basic integration problems.
// Basic helper functions for parsing and evaluating math expressions
// Note: This is a highly simplified parser for demonstration.
// A robust solution would require a dedicated math parsing library.
function parseFunction(funcString) {
// Replace common shorthand and normalize
funcString = funcString.toLowerCase().replace(/\s+/g, "); // Remove spaces
funcString = funcString.replace(/pi/g, Math.PI.toString());
funcString = funcString.replace(/e/g, Math.E.toString());
funcString = funcString.replace(/x\^(\d+)/g, 'Math.pow(x, $1)');
funcString = funcString.replace(/x(?!\^)/g, 'x'); // Ensure x is handled
funcString = funcString.replace(/\^/g, '**'); // Use JS power operator
funcString = funcString.replace(/sin/g, 'Math.sin');
funcString = funcString.replace(/cos/g, 'Math.cos');
funcString = funcString.replace(/tan/g, 'Math.tan');
funcString = funcString.replace(/log/g, 'Math.log'); // Natural log by default in JS Math.log
funcString = funcString.replace(/ln/g, 'Math.log'); // Alias for natural log
funcString = funcString.replace(/sqrt/g, 'Math.sqrt');
funcString = funcString.replace(/abs/g, 'Math.abs');
return funcString;
}
// Function to perform basic symbolic integration (very limited)
function symbolicIntegrate(funcString) {
funcString = funcString.trim();
if (!funcString) {
return "Error: Function cannot be empty.";
}
// Very basic polynomial integration: ax^n -> a*x^(n+1)/(n+1)
// This is a HUGE simplification and only handles very specific cases.
// A real symbolic integrator is extremely complex.
// Check for simple polynomial terms like ax^n
var polyMatch = funcString.match(/(-?\d*\.?\d*)\*?x\^(\d+)/);
if (polyMatch) {
var coefficient = parseFloat(polyMatch[1] || '1');
var exponent = parseInt(polyMatch[2]);
if (exponent === 0) { // Handle constants like 5
return (coefficient) + "*x + C";
}
var newExponent = exponent + 1;
var newCoefficient = coefficient / newExponent;
// Format output nicely
var sign = newCoefficient 0 && parts[0] !== ") {
coefficient = parseFloat(parts[0].replace('+', "));
}
if (!isNaN(coefficient)) {
var newExponent = 2;
var newCoefficient = coefficient / newExponent;
var sign = newCoefficient < 0 ? '-' : '';
var absCoeff = Math.abs(newCoefficient);
var coeffStr = (absCoeff === 1) ? '' : absCoeff.toString();
return sign + coeffStr + "x^" + newExponent + " + C";
}
}
// Check for constant terms (e.g., '5', '-3.14')
var constantMatch = funcString.match(/^([+-]?\d*\.?\d+)$/);
if (constantMatch) {
var constant = parseFloat(constantMatch[1]);
if (!isNaN(constant)) {
return constant + "*x + C";
}
}
// Very basic handling for sin(x), cos(x), e^x
if (funcString === 'sin(x)') return "-cos(x) + C";
if (funcString === 'cos(x)') return "sin(x) + C";
if (funcString === 'e^x') return "e^x + C";
if (funcString === '1/x') return "ln|x| + C";
// If no simple rule applies, return a message
return "Complex function or unsupported format. Cannot provide symbolic integral.";
}
function calculateIntegral() {
var functionInput = document.getElementById("functionInput").value;
var resultDiv = document.getElementById("result");
if (!functionInput) {
resultDiv.innerText = "Please enter a function.";
return;
}
try {
// For this calculator, we'll stick to a very basic symbolic approach
// as general symbolic integration is extremely complex.
var integralResult = symbolicIntegrate(functionInput);
resultDiv.innerText = integralResult;
} catch (error) {
console.error("Calculation error:", error);
resultDiv.innerText = "Error calculating integral. Check your input format.";
}
}