Antiderivative Calculator

What this calculator does

This calculator finds the antiderivative (also called the indefinite integral) of a polynomial function. You type an expression like 3x² + 2x − 5, and the tool returns the family of functions whose derivative equals what you typed, plus the constant of integration C. It also shows the work term by term, so you can copy the reasoning into homework, a tutorial, or notes — not just the final answer.

Scope: this tool is deliberately narrow

The calculator handles polynomial antiderivatives only: terms of the form a · xⁿ for any real coefficient a and integer power n. It applies three rules:

• Power rule: ∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C for n ≠ −1.
• Sum rule: the integral of a sum is the sum of the integrals.
• Constant multiple rule: constants pull through unchanged.

Plus the single non-polynomial special case ∫ a/x dx = a · ln|x| + C, because it shows up so often alongside polynomial terms that excluding it would be hostile.

What it does not handle: sin, cos, tan, eˣ, general logarithms, partial fractions, chain-rule substitution, integration by parts, trigonometric substitution, or anything that needs a symbolic algebra system. For those, use SymPy or Wolfram Alpha. The honesty matters — pretending to handle every integral and then crashing on sin(x) is worse than saying “polynomials only” up front.

How to read the result

The output looks like x³ + x² − 5x + C. The + C is the constant of integration, and it represents the fact that every antiderivative of a given function is determined only up to an additive constant. If you differentiate x³ + C you get 3x² regardless of what C is — so when running the reverse process, we keep C as an unknown.

The calculator shows the work in steps. For each input term, it states the power-rule application: increase the exponent by one, then divide by the new exponent. So ∫ 3x² dx becomes (3 / 3) · x³ = x³, and ∫ −5 dx becomes −5x (a constant integrates to “constant times x”). The 1/x case prints as ln|x| with the absolute-value bars, because the natural log is only defined for positive arguments.

Worked examples

Example 1: ∫ (3x² + 2x − 5) dx

Sum: x³ + x² − 5x + C.

Example 2: ∫ (1/x + x⁴) dx

Sum: x⁵/5 + ln|x| + C.

Example 3: ∫ 6 dx A bare constant integrates to itself times x, so the answer is 6x + C. This catches people out because there’s no x in the input — the answer “appears” to gain a variable. The power rule view: 6 = 6x⁰, so ∫ 6x⁰ dx = 6 · x¹ / 1 = 6x.

Indefinite vs definite integrals

This tool computes indefinite integrals — functions, not numbers. A definite integral ∫ₐᵇ f(x) dx evaluates the antiderivative at two endpoints and subtracts: F(b) − F(a). The + C cancels in that subtraction (it’s added to both sides and then differenced away), which is why definite integrals don’t carry a constant.

To use this tool for a definite integral, compute the indefinite form, then evaluate it at your endpoints by hand. Example: ∫₀² 3x² dx = [x³]₀² = 8 − 0 = 8.

Common mistakes to avoid

Forgetting + C on indefinite integrals. Lose points on every exam this way. Definite integrals don’t carry it; indefinite ones always do.

Dividing by n instead of n + 1. The power rule for integration divides by the new power, not the old. ∫ x⁴ dx = x⁵ / 5, not x⁵ / 4.

Trying to integrate by substitution mentally. ∫ (2x + 1)² dx is not (2x + 1)³ / 3 + C — that would require accounting for the chain rule, which this tool does not. Expand the bracket first: (2x + 1)² = 4x² + 4x + 1, which the calculator integrates term by term.

Privacy

This calculator parses your input and runs arithmetic in JavaScript on your device. There are no fetch calls, no analytics on the expression you type, no server-side logging.