Integral Calculator

Indefinite Integral
∫ x² dx = x³/3 + C
for f(x) = x²

📐 Symbolic Antiderivative

What Is Integration?

Integration is the reverse of differentiation. The indefinite integral (or antiderivative) of f(x) is a family of functions F(x) + C whose derivative is f(x). The definite integral from a to b is a specific number — the net signed area between the curve y = f(x) and the x-axis over that interval.

How to Use This Integral Calculator

  1. Enter your function of x, using * for multiplication and ^ for powers (e.g. x^2 + sin(x)).
  2. Choose Indefinite for a symbolic antiderivative, or Definite to also compute a numeric area between two bounds.
  3. For a definite integral, enter the lower bound a and upper bound b.
  4. Click Integrate to see the result.

The Power Rule for Integration

Power Rule (Integration)
∫ xⁿ dx = x⁽ⁿ⁺¹⁾ / (n + 1) + C (n ≠ −1)
n = the exponent (a constant)
C = the constant of integration

Worked Example

Example: ∫ x² dx

Add 1 to the exponent: x³. Divide by the new exponent: x³/3.

Result: ∫ x² dx = x³/3 + C

How Simpson's Rule Works

For a definite integral, this calculator uses Simpson's rule — a numeric method that approximates the curve between points using small parabolic segments instead of straight lines, which makes it far more accurate than simpler methods like the trapezoidal rule. It splits [a, b] into a large, even number of subintervals of width h and combines the sample values with alternating weights:

Simpson's Rule
∫ₐᵇ f(x) dx ≈ (h/3)·[f(a) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + f(b)]
h = (b − a) / n, where n is the (even) number of subintervals

This tool always uses a large number of subintervals (1,000+), so the numeric result is accurate to many decimal places for smooth functions — even ones with no elementary antiderivative at all.

Common Mistakes to Avoid

  • Forgetting the "+ C" on an indefinite integral — without it, you've only found one antiderivative out of infinitely many.
  • Using the power rule ∫xⁿdx = x⁽ⁿ⁺¹⁾/(n+1) when n = −1, which requires the special case ∫x⁻¹dx = ln|x| + C instead.
  • Assuming every function has a "nice" elementary antiderivative — many common functions (like e^(−x²)) simply don't.

Frequently Asked Questions

Symbolic integration finds an exact formula (the antiderivative) using algebraic rules. Numeric integration instead estimates the value of a definite integral by sampling the function at many points — it doesn't give you a formula, but it works even when no formula exists.

Functions like e^(−x²) or sin(x)/x provably cannot be expressed as a finite combination of polynomials, exponentials, logarithms, and trig functions — even though their derivatives are perfectly ordinary. This calculator will say so plainly instead of guessing.

Very accurate for smooth, well-behaved functions — error shrinks with the fourth power of the interval width. With over 1,000 subintervals, results are typically accurate to at least 6–8 decimal places, except very close to a genuine discontinuity or asymptote inside the interval.

C represents an arbitrary constant. Since the derivative of any constant is zero, every antiderivative of f(x) can have any constant added to it and still be a valid antiderivative — "+C" acknowledges this whole family of solutions.

Not currently — the definite integral mode requires two finite bounds. Improper integrals with infinite limits or interior discontinuities need special handling that's outside this tool's current scope.