Integral Step-by-Step Calculator
Solve polynomial indefinite and definite integrals with detailed steps.
Format: ax^n + bx + c (Use '^' for exponents)
Understanding Integration: The Power Rule
Integration is the fundamental process of finding the antiderivative of a function. For polynomials, we primarily use the Power Rule for Integration, which is the inverse of the power rule for derivatives.
The general formula for the power rule is:
∫ xn dx = (xn+1) / (n+1) + C
How to Use This Calculator
- Enter your function: Input the polynomial expression using standard notation (e.g.,
4x^3). - Set Bounds: If you leave the bounds empty, the calculator provides the indefinite integral (the antiderivative + C). If you provide bounds, it calculates the area under the curve (definite integral).
- Review Steps: The calculator breaks down each term, showing how the exponent is increased and how the coefficient is adjusted.
Example Calculation
Problem: Calculate the definite integral of 3x^2 + 2x from 0 to 2.
Step 1: Find the antiderivative.
∫ 3x^2 dx = (3/3)x^3 = x^3
∫ 2x dx = (2/2)x^2 = x^2
Antiderivative: x^3 + x^2
Step 2: Evaluate at Upper Bound (2).
(2)^3 + (2)^2 = 8 + 4 = 12
Step 3: Evaluate at Lower Bound (0).
(0)^3 + (0)^2 = 0
Result: 12 – 0 = 12.
Rules for Integration
- Constant Rule: The integral of a constant
kiskx + C. - Sum/Difference Rule: The integral of a sum is the sum of the integrals.
- Definite Integral: Calculated using the Fundamental Theorem of Calculus: ∫ab f(x)dx = F(b) – F(a).
Integration Steps:
"; var antiderivativeFunc = []; // Array of objects for numeric calculation for (var i = 0; i = 0 && i > 0) termStr += " + "; if (simplifiedCoeff === -1) termStr += "-"; else if (simplifiedCoeff !== 1 || newExp === 0) termStr += simplifiedCoeff; if (newExp === 1) termStr += "x"; else if (newExp !== 0) termStr += "x^" + newExp; antiderivativeTerms.push(termStr); antiderivativeFunc.push({ c: newCoeff, e: newExp }); stepsHtml += "Term: " + term + " → Increase exponent to " + newExp + ", Divide coefficient " + coeff + " by " + newExp + " = " + simplifiedCoeff + "x^" + newExp + "
";
}
var fullAntiderivative = antiderivativeTerms.join("") + " + C";
indefiniteDiv.innerHTML = "Indefinite Integral:∫ f(x)dx = " + fullAntiderivative + "
";
stepsDiv.innerHTML = stepsHtml;
// Handle Definite Integral
if (lower !== "" && upper !== "") {
var lVal = parseFloat(lower);
var uVal = parseFloat(upper);
var evalUpper = 0;
var evalLower = 0;
for (var j = 0; j < antiderivativeFunc.length; j++) {
evalUpper += antiderivativeFunc[j].c * Math.pow(uVal, antiderivativeFunc[j].e);
evalLower += antiderivativeFunc[j].c * Math.pow(lVal, antiderivativeFunc[j].e);
}
var finalResult = evalUpper – evalLower;
definiteDiv.innerHTML = "