Understanding antiderivatives, also known as indefinite integrals, is a fundamental concept in calculus. An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). In simpler terms, it's the reverse process of differentiation.
When finding an antiderivative, we always add a constant of integration, denoted as + C. This is because the derivative of any constant is zero, meaning that infinitely many functions can have the same derivative, differing only by a constant value.
The Power Rule for Integration
One of the most common rules for finding antiderivatives is the Power Rule. It states that for a function of the form f(x) = axn, where a is a coefficient and n is an exponent, its antiderivative F(x) is given by:
F(x) = (a / (n + 1)) * x(n + 1) + C
This rule applies to all real numbers n, except for n = -1. When n = -1, the function is f(x) = a/x, and its antiderivative involves the natural logarithm:
F(x) = a * ln|x| + C
How to Use This Calculator
This calculator helps you find the antiderivative of a single monomial term in the form axn. Simply enter the coefficient a and the exponent n into the fields below, and the calculator will provide the antiderivative along with the steps involved.
Antiderivative of a Monomial Calculator
Enter the numerical coefficient of the term (e.g., 2 for 2x^3).
Enter the exponent of 'x' (e.g., 3 for 2x^3).
Results:
Steps:
'; if (exponentN === -1) { // Special case for n = -1 (1/x) var finalCoeff = coefficientA; antiderivativeString = finalCoeff + ' ln|x| + C'; stepsHtml += '1. Identify the function asf(x) = ' + formatTerm(coefficientA, exponentN) + '.';
stepsHtml += '2. Recognize that the exponent n = -1. For functions of the form a/x, the antiderivative rule is ∫(a/x) dx = a * ln|x| + C.';
stepsHtml += '3. Apply the rule: F(x) = ' + finalCoeff + ' ln|x| + C.';
} else {
// General Power Rule
var newExponent = exponentN + 1;
var newCoefficient = coefficientA / newExponent;
// Format the new coefficient for display (fraction or decimal)
var formattedNewCoeff;
if (newCoefficient === Math.floor(newCoefficient)) {
formattedNewCoeff = newCoefficient.toString(); // Integer
} else {
var tolerance = 1.0E-6; // For floating point comparison
var foundFraction = false;
for (var d = 1; d <= 100; d++) { // Check denominators up to 100
var n_approx = newCoefficient * d;
if (Math.abs(n_approx – Math.round(n_approx)) < tolerance) {
formattedNewCoeff = Math.round(n_approx) + '/' + d;
foundFraction = true;
break;
}
}
if (!foundFraction) {
formattedNewCoeff = newCoefficient.toFixed(4); // Fallback to decimal
}
}
var formattedXTerm = '';
if (newExponent === 0) {
formattedXTerm = ''; // x^0 = 1, but this case should not be reached if n != -1
} else if (newExponent === 1) {
formattedXTerm = 'x';
} else {
formattedXTerm = 'x' + newExponent + '';
}
if (newCoefficient === 1 && newExponent !== 0) {
antiderivativeString = formattedXTerm + ' + C';
} else if (newCoefficient === -1 && newExponent !== 0) {
antiderivativeString = '-' + formattedXTerm + ' + C';
} else if (newCoefficient === 0) {
antiderivativeString = 'C'; // If original coefficient was 0
}
else {
antiderivativeString = formattedNewCoeff + formattedXTerm + ' + C';
}
stepsHtml += '1. Identify the function as f(x) = ' + formatTerm(coefficientA, exponentN) + '.';
stepsHtml += '2. Apply the Power Rule for Integration: ∫xn dx = (xn+1)/(n+1) + C.';
stepsHtml += '3. For f(x) = ' + coefficientA + 'x' + exponentN + ', we have:';
stepsHtml += '∫' + coefficientA + 'x' + exponentN + ' dx = ' + coefficientA + ' * (x(' + exponentN + ')+1)/(' + exponentN + '+1) + C';
stepsHtml += '4. Simplify the exponent and coefficient:';
stepsHtml += 'New Exponent: ' + exponentN + ' + 1 = ' + newExponent + '';
stepsHtml += 'New Coefficient: ' + coefficientA + ' / ' + newExponent + ' = ' + formattedNewCoeff + '';
stepsHtml += '5. Combine to get the antiderivative: F(x) = ' + antiderivativeString + '.';
}
antiderivativeResultDiv.innerHTML = 'Antiderivative F(x): ' + antiderivativeString;
stepsOutputDiv.innerHTML = stepsHtml;
}
// Run calculation on page load with default values
document.addEventListener('DOMContentLoaded', calculateAntiderivative);