General Antiderivative Calculator
An antiderivative is a function whose derivative gives you back the function you started with: the indefinite integral, plus the constant of integration. Pick one of the four function families below, enter the numbers, and this calculator returns the antiderivative, names the rule it used, walks through the steps, and verifies the answer by differentiating it back.
Scope, stated plainly: this tool handles one term from one of the four families above. It does not split sums, undo the chain rule, or integrate products; the article below shows what to do for those instead.
Quick Answer
For the pre-filled example, the general antiderivative of 6x^2 is 2.0000x^3 + C: divide the coefficient by the new exponent (6 / 3 = 2), raise the power by one, and add the constant of integration.
The tool covers four families with the same pattern: powers use the power rule (with n = -1 switched to ln|x|), e^x is its own antiderivative, four trig functions use the standard trig rules, and ln(x) gives x ln(x) – x by parts. Every result is checked by differentiating it back inside the steps panel.
- Quick Answer
- How to use the general antiderivative calculator
- The four rules this calculator uses
- Worked example: the antiderivative of 6x^2
- More runs, printed exactly as the tool prints them
- What this calculator cannot do, stated honestly
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the general antiderivative calculator
Pick the family, type the numbers, and click Calculate: the result box shows your function, its antiderivative, the rule that was applied, and a step list ending in a differentiation check.
- Choose the function type. Power for a·x^n, Exponential for a·e^x, Trigonometric for sin, cos, sec² or csc², and Logarithmic for a·ln(x). The form swaps to the fields that family needs.
- Enter the coefficient a (and the exponent n for powers, or the trig function for trig mode). The fields start pre-filled with the worked examples from this article: 6 and 2 for power, 3 for exponential, 2 sin(x) for trig, 3 for logarithmic.
- Click Calculate. Nothing is computed until you do; there is no auto-calculate while you type.
- Read all four lines. The antiderivative is the headline, but the rule line tells you which pattern to learn, and the steps line ends with the proof: differentiating the answer returns your function.
- Copy or share from the buttons inside the results box. Reset restores the worked-example values and hides the results.
One input rule applies everywhere: the coefficient and exponent fields must contain real numbers. Leave one blank and the tool answers “Enter valid numbers first.” instead of guessing, which is the right behaviour for a tool students copy answers from. If your function is a sum such as x^2 + 3x, run each term separately and add the results; the antidifferentiation calculator covers the concept from the theory side.
The four rules this calculator uses
Every answer comes from one of four rules, plus the constant multiple rule that pulls the coefficient a out front first: the integral of a·f(x) is a times the integral of f(x).
| Family | You enter | Antiderivative | Rule |
|---|---|---|---|
| Power | a·x^n, n not -1 | a·x^(n+1)/(n+1) + C | Power rule |
| Power, exception | a/x (n = -1) | a ln|x| + C | Logarithm rule |
| Exponential | a·e^x | a·e^x + C | e^x is its own derivative |
| Trigonometric | a sin(x) | -a cos(x) + C | Derivative of cos is -sin |
| Trigonometric | a cos(x) | a sin(x) + C | Derivative of sin is cos |
| Trigonometric | a sec²(x) | a tan(x) + C | Derivative of tan is sec² |
| Trigonometric | a csc²(x) | -a cot(x) + C | Derivative of cot is -csc² |
| Logarithmic | a·ln(x) | a(x ln(x) – x) + C | Integration by parts |
The power rule is the workhorse: add 1 to the exponent, divide by that new exponent. It works for negative exponents (the antiderivative of x^-2 is -x^-1) and fractional ones (the antiderivative of x^0.5 is x^1.5/1.5, printed as 0.6667x^1.5 when a = 1). The single place it breaks is n = -1, where the new exponent is 0 and the division is by zero; calculus fills that hole with ln|x|, and this calculator switches to it automatically. The trig signs are the classic trap: sin integrates to -cos, and csc² integrates to -cot, while cos and sec² keep a positive sign.
Key takeaways
- An antiderivative is a family, not a single function: the + C is part of the answer, and “general antiderivative” means the version with C included.
- The power rule covers every exponent except -1; that one case belongs to ln|x|.
- e^x and ln(x) are the two special shapes: one reproduces itself, the other needs integration by parts.
- Any antiderivative can be checked in seconds by differentiating it, and this tool prints that check on every run.
Worked example: the antiderivative of 6x^2
The pre-filled power example runs in four moves: pull the constant out, apply the power rule, simplify, then verify by differentiating.
6·x^2 becomes 6 times the integral of x^2.
x^2 becomes x^3/3: exponent up by one, divide by the new exponent.
6 · x^3/3 = 2x^3, and the tool prints the coefficient to four decimals: 2.0000x^3 + C.
Differentiating 2x^3 gives 6x^2, the function you started with. Check passed.
The same rhythm handles the other families. For 3e^x there is almost nothing to do: e^x differentiates to itself, so the antiderivative is 3e^x + C with the coefficient untouched. For 3 ln(x), integration by parts with u = ln(x) and dv = dx gives x ln(x) minus the integral of x · (1/x), which is just x, so the answer is 3(x ln(x) – x) + C. Differentiating that returns 3(ln(x) + 1 – 1) = 3 ln(x), which is the verification line the tool prints.
More runs, printed exactly as the tool prints them
Each row below was produced by this calculator and hand-checked by differentiating the answer. Use them to confirm the tool on your own numbers before you trust it on homework.
| Mode and input | Tool output | Differentiate it back |
|---|---|---|
| Power, a = 6, n = 2 | 2.0000x^3 + C | 3 · 2x^2 = 6x^2 |
| Power, a = 1, n = 2 | 0.3333x^3 + C | 3 · (1/3)x^2 = x^2 (coefficient shown rounded) |
| Power, a = 6, n = -1 | 6 ln|x| + C | 6 · (1/x) = 6/x |
| Power, a = 3, n = -2 | -3.0000x^-1 + C | -1 · -3x^-2 = 3x^-2 |
| Exponential, a = 3 | 3e^x + C | 3e^x, unchanged |
| Trig, 2 sin(x) | -2 cos(x) + C | -2 · -sin(x) = 2 sin(x) |
| Trig, 2 sec²(x) | 2 tan(x) + C | 2 sec²(x) |
| Logarithmic, a = 3 | 3(x ln(x) – x) + C | 3(ln(x) + 1 – 1) = 3 ln(x) |
What this calculator cannot do, stated honestly
It handles one term from one family at a time, and nothing else. Three common homework shapes fall outside that scope, and no setting inside the tool will fix them.
Sums and differences such as x^2 + 3x – 5: the tool has no way to enter a second term. The mathematics is friendly here, since antiderivatives split over sums, so run each term separately (x^2 gives 0.3333x^3, 3x gives 1.5000x^2, the constant 5 is a power with n = 0 and gives 5.0000x^1) and add the pieces, keeping a single + C at the end.
Composite functions such as sin(3x) or (2x + 1)^4 need the chain rule run in reverse (u-substitution), and products such as x·e^x need integration by parts. This tool does neither, because its input is a coefficient and an exponent, not a typed expression. A full symbolic solver is the right instrument for those; pretending a four-family picker handles them would be a false claim, so the limit is printed here and under the calculator itself.
Two smaller quirks are worth knowing. In Power mode the new coefficient is printed to four decimal places, so exact answers like (1/3)x^3 appear as 0.3333x^3; the exponent and the rule are exact, only that one printed number is rounded. And in Trig mode the sign is built into the rule (sin gives -cos), so a negative coefficient there prints as a double minus, for example –3 cos(x) + C; enter a positive coefficient and read the sign from the rule instead.
What competing calculators miss
Most rivals go wide: type any expression and a symbolic engine answers. That breadth costs exactly the things a student learning antiderivatives needs, namely a named rule, a short step list, and a stated scope.
Symbolab and Pearson’s integral calculator accept free-form input and handle substitution and parts cases this page refuses, but the antiderivative becomes one operation among dozens, with step trees built for every calculus problem rather than this one. EduInput’s antiderivative calculator takes typed expressions too, and its guidance amounts to checking your syntax. BraveCalculator’s step-by-step integral tool teaches well, including a by-parts example, yet it never proves its final answer by differentiating it. IntegralCalculator.app is admirably honest about supporting only classroom patterns, but it stops at a draft result without a verification line. Firgell’s indefinite integral tool is the closest cousin, a coefficient-and-exponent picker over the same four families, and it still presents its formulas in the article rather than naming the rule inside the result box. This calculator’s narrower bet: four families, the rule named on your numbers, three steps, and a differentiation check printed on every run.
Frequently asked questions
An antiderivative of f(x) is any function F(x) whose derivative equals f(x). It is the indefinite integral: the integral of f(x) dx = F(x) + C, where C is the constant of integration. This calculator returns one such F(x) for each of its four function families and checks it by differentiating it back.
Because constants vanish when you differentiate. x^3, x^3 + 5, and x^3 – 100 all differentiate to 3x^2, so the antiderivative of 3x^2 is a whole family of functions, and + C stands for all of them. The word general in this calculator’s name refers to exactly that: the general antiderivative is the version with the constant included.
Applying it would give x^0/0, a division by zero. The integral of 1/x is instead ln|x| + C, which you can confirm because the derivative of ln|x| is 1/x. This calculator detects n = -1 in Power mode and switches to the logarithm rule automatically.
Four families, one term at a time: a power term (a times x^n) for any n except -1, which uses the log rule; an exponential (a times e^x); four trig functions (sin, cos, sec^2, csc^2) with a coefficient; and a logarithm (a times ln(x)). It cannot split sums like x^2 + 3x, undo the chain rule in sin(3x), or integrate products like x times e^x; those need term-by-term work by hand or a full symbolic solver.
It differentiates the result. If differentiating F(x) gives back your original f(x), the antiderivative is correct, and the steps panel prints that check as its last line for every calculation. It is the same test you can run on any antiderivative by hand, and it is the fastest way to catch a sign error in the trig rules.
ln|x| + C. The absolute value matters: ln(x) alone is only defined for positive x, while 1/x is defined for negative x too, and ln|x| differentiates to 1/x on both sides of zero. Enter coefficient 1 and exponent -1 in Power mode to see it.
The tool prints the new coefficient, a divided by (n+1), to four decimal places, so the antiderivative of x^2 shows as 0.3333x^3 + C instead of the exact (1/3)x^3 + C. The rule and the exponent are exact; only the printed coefficient is rounded, and the verification step uses that same rounded value. For exact fraction work, do that one division by hand.
x ln(x) – x + C, found by integration by parts. With a coefficient a it becomes a(x ln(x) – x) + C, which is what this calculator returns in Logarithmic mode: enter a = 3 and you get 3(x ln(x) – x) + C. Differentiating it gives ln(x) + 1 – 1 inside the brackets, which collapses back to ln(x).
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References
- Weisstein, Eric W., Wolfram MathWorld. Antiderivative. Accessed October 21, 2026.
- Weisstein, Eric W., Wolfram MathWorld. Indefinite Integral. Accessed October 21, 2026.
- Dawkins, Paul, Lamar University, Paul’s Online Math Notes. Indefinite Integrals. Accessed October 21, 2026.
- Dawkins, Paul, Lamar University, Paul’s Online Math Notes. Computing Indefinite Integrals. Accessed October 21, 2026.
- Massachusetts Institute of Technology, MIT OpenCourseWare. 18.01 Single Variable Calculus, Fall 2006. Accessed October 21, 2026.
- OpenStax, Rice University. Calculus Volume 1, Section 4.10 (Antiderivatives). Accessed October 21, 2026.