Antidifferentiation Calculator
Antidifferentiation is differentiation run in reverse: you are handed a rate, a slope, or a growth rule, and you need the function it came from. The Antidifferentiation Calculator below covers the four families that make up most first-year calculus work: a power term a x^n, an exponential a e^x, a trigonometric term (sin, cos, or sec squared), and a logarithm a ln(x). Pick the family, enter the coefficient and the details, and the tool returns the antiderivative with the working shown step by step, not just the final line.
The scope is deliberate. This is not a general computer algebra system and it does not pretend to parse any expression you type. It handles the forms it lists, handles them exactly, and shows its reasoning so you can check it against your own work. That includes the case simple tools get wrong: x to the power of negative 1, where the power rule divides by zero and the correct answer is a logarithm.
Scope: this calculator integrates one term from the four families listed. Sums of terms are integrated one term at a time (integrate each term, then add the results). Products, quotients, and composite functions need substitution or parts by hand, and the article below explains where this tool stops.
Quick Answer
Key takeaways
- An antiderivative is a family, not a single function: every answer carries a + C because constants vanish when differentiated.
- The power rule (add 1 to the power, divide by the new power) works for every power except n = -1. That one case is ln|x|, and tools that miss it divide by zero.
- Trig and exponential integrals are derivative pairs read backward. The only sign that traps students is sine: its integral is negative cosine.
- ln(x) is the parts case in this tool: its antiderivative is x ln(x) minus x, and differentiating that result is a clean way to verify it.
- Quick Answer
- What antidifferentiation is, and why the + C is not decoration
- The four rules this calculator uses
- Worked example: integrating 3x^2 by hand and by tool
- The n = -1 exception, and the mistakes that surround it
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
What antidifferentiation is, and why the + C is not decoration
Differentiation takes a function and returns its rate of change. Antidifferentiation starts from the rate and recovers the function. If F'(x) = f(x), then F is an antiderivative of f, and the indefinite integral is written as the integral of f(x) dx = F(x) + C, the notation OpenStax Calculus, section 4.10, on antiderivatives uses to define the most general antiderivative.
The constant exists because differentiation deletes information. x^3 + 7 and x^3 minus 4 both differentiate to 3x^2, so no process working backward from 3x^2 can tell you which constant you started with. An initial condition settles it: if the problem also says F(0) = 7, the constant is 7, and the calculator’s C field lets you pin that specific value in place of the symbolic + C. Everything else in integration builds on this recovery idea, including finding the length of a curve, where an arc length calculator applies an antiderivative to a square-root integrand.
The four rules this calculator uses
Each family in the tool maps to one rule, and the steps panel names the rule and shows its arithmetic. Nothing is pattern-matched behind the scenes beyond these four.
Identify the family. Is the term a power of x, e^x, a trig function, or ln(x)? The family chooses the rule. Most errors happen here, by forcing the power rule onto a form it does not fit.
Apply the rule with its arithmetic visible. For powers: new power, then new coefficient. For pairs: match the derivative you know. For ln(x): run the parts setup, u = ln(x), dv = dx.
Add the constant and differentiate to check. The derivative of your answer must return the original function exactly. This ten-second check, recommended in Paul’s Online Math Notes on indefinite integrals, catches nearly every sign and coefficient slip.
Worked example: integrating 3x^2 by hand and by tool
The calculator is pre-filled with f(x) = 3x^2 and C = 0, and it returns F(x) = x^3 + C through these steps: the power rises from 2 to 3, the coefficient 3 is divided by the new power 3 to give 1, and the constant is attached. By hand you would write the same three lines. The example is simple on purpose: it isolates the power rule’s two moves so the division step, the one students most often skip, stays visible.
The table below collects every form the tool supports, each one hand-worked and verified by differentiating the answer. Use it as the reference card for what the calculator will return before you press the button.
| Integrand | Antiderivative | Rule, and the check |
|---|---|---|
| 3x^2 | x^3 + C | Power rule (3 / 3 = 1). Derivative of x^3 is 3x^2. |
| 5x^4 | x^5 + C | Power rule (5 / 5 = 1). Derivative of x^5 is 5x^4. |
| 6x^-2 | -6x^-1 + C | Power rule (6 / -1 = -6). Derivative of -6x^-1 is 6x^-2. |
| 2x^-1, that is 2/x | 2 ln|x| + C | The exception: power rule would divide by 0. Derivative of 2 ln|x| is 2/x. |
| 4e^x | 4e^x + C | e^x is its own derivative and its own integral. |
| 2 sin(x) | -2 cos(x) + C | Pair with sign flip. Derivative of -2 cos(x) is 2 sin(x). |
| 3 cos(x) | 3 sin(x) + C | Pair. Derivative of 3 sin(x) is 3 cos(x). |
| sec^2(x) | tan(x) + C | Pair. Derivative of tan(x) is sec^2(x). |
| 2 ln(x) | 2(x ln(x) – x) + C | Parts. Derivative of x ln(x) – x is ln(x) + 1 – 1 = ln(x). |
Sums are handled term by term, because the integral of a sum is the sum of the integrals: to integrate 3x^2 + 4e^x, run the tool twice and add the results, x^3 + 4e^x + C, using one shared constant. The same term-by-term logic underlies area problems between curves, which the area between two functions calculator applies with definite integrals.
The n = -1 exception, and the mistakes that surround it
One power breaks the power rule. At n = -1 the new power is 0 and the new coefficient is a divided by 0, which is undefined, not zero and not infinity. The correct antiderivative of 1/x is ln|x|, because the derivative of ln|x| is 1/x on both sides of zero, which is also why the absolute value bars belong there. OpenStax states the exception alongside the rule itself, and Khan Academy’s integration rules summary lists the 1/x case separately for the same reason. This calculator routes n = -1 to the logarithm automatically and says so in the steps; the previous version of this tool returned an infinite coefficient, a defect this rewrite fixes and documents.
Three other errors account for most wrong homework answers in this territory. First, dividing by the old power instead of the new one: the integral of 3x^2 is not (3/2)x^3. Second, dropping the sign on sine: the integral of sin(x) is -cos(x), and the check by differentiation exposes the error instantly. Third, treating ln(x) as if its integral were 1/x: 1/x is the derivative of ln(x), and reversing a derivative pair in the wrong direction is the single most common ln mistake. When your work moves past these forms into composite functions, substitution is the next technique, and inverse trig results such as the integral of 1/(1 + x^2) giving arctan(x) connect to the arctan calculator.
What competing calculators miss
The free-text solvers at the top of this search are powerful, and for a composite or product integrand you should use one. For the foundational forms, though, they trade away three things this page keeps.
Winning move 1: steps that mirror the classroom rules. The big solvers show steps too, but wrapped around a general engine that may choose a substitution detour for a plain power term. Here every step is the rule itself, with the arithmetic (3 / 3 = 1) printed, so the working matches what a teacher marks.
Winning move 2: the n = -1 case handled, and the failure mode named. Form-based rival tools that apply the power rule blindly return a division by zero dressed up as a number. This tool detects the exception, returns a ln|x| + C, and explains in the steps why the rule cannot apply. Our hand-verified table above proves each supported form by differentiating it back.
Winning move 3: honest scope instead of silent misparsing. Free-text entry fails quietly on implicit multiplication and ambiguous notation, returning a confident wrong answer. A form-based tool cannot misread its own menu: four families, exact rules, and a printed statement of where the tool stops (products, quotients, composites) and what to learn next.
Frequently asked questions
An antiderivative is a function: F is an antiderivative of f when F'(x) = f(x). The indefinite integral is the notation for the whole family of antiderivatives, written as the integral of f(x) dx = F(x) + C. In practice the words are used interchangeably, and this calculator returns the family, constant included.
Because the derivative of any constant is 0, functions that differ only by a constant share the same derivative, so working backward cannot recover which constant you started from. The + C records that whole family honestly. An initial condition, such as F(0) = 7, is what pins one specific value down.
ln|x| + C. The power rule cannot produce it: at n = -1 the rule divides the coefficient by n + 1 = 0. The logarithm fills the gap because the derivative of ln|x| is 1/x for positive and negative x alike, which is why the absolute value is part of the answer.
Term by term. The integral of a sum is the sum of the integrals, so integrate 3x^2 to get x^3, integrate 4e^x to get 4e^x, and combine with a single constant: x^3 + 4e^x + C. Run this calculator once per term and add the results.
x ln(x) – x + C, found by integration by parts with u = ln(x) and dv = dx. The check is quick: differentiating x ln(x) – x gives ln(x) + 1 – 1, which is ln(x). A common error is answering 1/x, but that is the derivative of ln(x), not its integral.
Because the derivative of cos(x) is -sin(x). Running that pair backward, the function whose derivative is sin(x) must be -cos(x): differentiate -cos(x) and the two negatives cancel to give sin(x). Forgetting this sign is the most frequent trig integration error.
No, and it says so on purpose. It integrates single terms from four families: powers, e^x, sin/cos/sec squared, and ln(x). Products such as x e^x need integration by parts set up by hand, composites need substitution, and definite integrals need the antiderivative evaluated at two bounds. A full symbolic solver is the right tool for those.
Differentiate it. The derivative of a correct antiderivative returns the original function exactly, constants and all. This is the most reliable habit in integration because it needs no answer key: if the derivative does not match, the error is in the step where the two sides first diverge, usually a coefficient or a sign.
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References
- OpenStax, Rice University, “4.10 Antiderivatives, Calculus Volume 1” – the definition of the most general antiderivative, the power rule and its n = -1 exception, and the basic integration formulas. OpenStax: Antiderivatives, Calculus Volume 1. Accessed October 2026.
- OpenStax, Rice University, “5.5 Substitution, Calculus Volume 1” – the next technique beyond this tool’s scope, for composite integrands. OpenStax: Substitution, Calculus Volume 1. Accessed October 2026.
- Paul Dawkins, Lamar University, “Indefinite Integrals, Paul’s Online Math Notes” – basic integration formulas and the differentiate-to-check verification method. Paul’s Online Math Notes: Indefinite Integrals. Accessed October 2026.
- Khan Academy, “Integration Rules” (AP Calculus AB) – the summary of common integration rules including the separate 1/x case. Khan Academy: Integration Rules. Accessed October 2026.
- Wolfram MathWorld, “Antiderivative” – the reference definition of an antiderivative and its relation to the indefinite integral. Wolfram MathWorld: Antiderivative. Accessed October 2026.
- MIT OpenCourseWare, “18.01 Single Variable Calculus, Fall 2006” – lecture course covering antiderivatives and the Fundamental Theorem of Calculus. MIT OpenCourseWare: Single Variable Calculus. Accessed October 2026.
- AnalyzeMath, “Integral Calculator with Examples” – step-by-step indefinite integral examples using the same differentiate-to-verify approach. AnalyzeMath: Integral Calculator. Accessed October 2026.