SamCalculator

Antiderivative Calculator

Find the indefinite integral of mathematical functions with step-by-step solutions, graph visualization, derivative verification, and detailed calculus explanations.

Function to integrate

∫dx

Common integral formulas

Power & constant

∫ k dx = k·x+c
∫ xⁿ dx = xn+1n+1+c
∫ (1/x) dx = ln(|x|)+c
∫ √x dx = 2x3/23+c

Exponential & logarithmic

∫ eˣ dx = ex+c
∫ aˣ dx = axln(a)+c
∫ ln x dx = x·ln(x)−x+c
∫ e^(kx) dx = ek·xk+c

Trigonometric

∫ sin x dx = −cos(x)+c
∫ cos x dx = sin(x)+c
∫ sec²x dx = tan(x)+c
∫ tan x dx = −ln(|cos(x)|)+c

Inverse-trig & hyperbolic

∫ 1/(1+x²) dx = arctan(x)+c
∫ 1/√(1−x²) dx = arcsin(x)+c
∫ sinh x dx = cosh(x)+c
∫ cosh x dx = sinh(x)+c

What Is An Antiderivative Calculator?

An antiderivative calculator finds the indefinite integral of a function — that is, a function F(x) whose derivative is the function f(x) you entered. Where a derivative tells you the instantaneous rate of change, an antiderivative runs the process in reverse: it reconstructs the original quantity from its rate. This tool parses your expression with a built-in symbolic engine, applies the correct integration rule, simplifies the result, verifies it by differentiating the answer back, plots both curves, and explains every step in plain English.

It handles polynomials, rational functions, exponentials, logarithms, all six trigonometric and inverse-trigonometric functions, hyperbolic functions, square roots, and many composite expressions through substitution and integration by parts. Every answer includes the mandatory constant of integration + C. It pairs naturally with the scientific calculator, the quadratic formula calculator, and the log calculator.

How Antiderivatives Work

Reverse of differentiation

F(x) is an antiderivative of f(x) if F′(x) = f(x). Integration undoes differentiation, so knowing the derivative rules backwards gives you the integration rules. That is why this calculator verifies its own answer by differentiating it.

A whole family of curves

Because the derivative of any constant is zero, if F(x) works then so does F(x) + 5, F(x) − 200, and F(x) + C for any constant C. The indefinite integral therefore represents an infinite family of parallel curves, all with the same slope at every x.

The +C is not optional

Leaving out the constant of integration is the single most common mistake. Without + C you have named one antiderivative instead of all of them, which matters the moment you apply an initial condition to pin down the specific curve.

Slope equals height

The defining geometric picture: at every point, the slope of the antiderivative F(x) equals the height of the original function f(x). Where f is positive, F rises; where f is zero, F is momentarily flat; where f is negative, F falls.

6 Ways To Use This Calculator

1

Check your homework

Integrate a function, then compare the answer and every step against your own working. The verification badge tells you instantly whether the result differentiates back to the original.

2

Learn the integration rules

The rule identifier names exactly which technique was used — power rule, u-substitution, integration by parts, partial fractions — and explains why it applies to your specific function.

3

Visualise f and F together

The interactive graph overlays your function and its antiderivative so you can see the slope-equals-height relationship. Drag to pan, zoom to inspect, toggle each curve on or off.

4

Export clean notation

Copy the answer as plain text, LaTeX for your report, or MathML for the web. The shareable link reopens the calculator with your exact function pre-loaded.

5

Explore what-if changes

Switch the integration variable, toggle simplification, or turn verification on and off — the result updates instantly so you can compare forms of the same answer.

6

Build calculus intuition

Try eˣ, sin x, and 1/x to memorise the special antiderivatives, then combine them into sums and products to watch linearity and integration by parts in action.

Best Practices For Integration

Simplify before you integrate, not after. Expanding a product, splitting a fraction into separate terms, or cancelling a common factor often converts an intimidating integral into a sum of textbook cases. This calculator simplifies internally, but on paper the habit saves you from reaching for substitution when the power rule would have done the job.

Choose your method by the shape of the integrand. A single power of the variable is the power rule; a composite function whose inner derivative also appears is a substitution; a product of two unlike functions is usually integration by parts (use the LIATE order — logarithmic, inverse-trig, algebraic, trigonometric, exponential — to pick the part to differentiate); a proper rational function calls for partial fractions.

Always verify by differentiating. Because integration and differentiation are inverses, taking the derivative of your answer should return the original integrand exactly. It is the fastest, most reliable check in all of calculus — and the reason this tool performs it automatically on every result.

Why Antiderivatives Matter

Physics and motion

Integrating acceleration gives velocity; integrating velocity gives position. Antiderivatives recover the whole history of a moving object from its rate of change, which is why they open every mechanics course.

Area and accumulation

The Fundamental Theorem of Calculus links the antiderivative to the area under a curve. Total distance, accumulated cost, delivered charge, and consumed fuel are all definite integrals evaluated from an antiderivative.

Probability and statistics

A probability density function integrates to a cumulative distribution function. Expected values, variances, and the normalising constants of continuous distributions all rely on finding antiderivatives.

Engineering and economics

Consumer and producer surplus, present value of a continuous income stream, work done by a variable force, and the response of an electrical circuit are computed by integrating a rate over an interval.

Where Integration Gets Tricky

The n = −1 exception

The power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) breaks at n = −1 because you would divide by zero. That single case, ∫1/x dx, is the natural logarithm ln|x| — the reason logarithms appear throughout integration.

No elementary antiderivative

Some perfectly ordinary functions — eˣ², sin(x)/x, √(1 + x³) — have no antiderivative expressible in elementary functions. This calculator says so honestly and points you to a definite (numerical) integral instead of inventing a wrong answer.

Absolute values in logs

The antiderivative of 1/x is ln|x|, not ln(x). The absolute value keeps the answer valid for negative inputs. This calculator always writes the |·| where it belongs so your domain stays correct.

Constants hiding as variables

Only the variable you integrate against is treated as a variable; every other letter is a constant that comes along for the ride. Choosing the wrong variable of integration silently changes the whole answer, so the dropdown makes it explicit.

Core Integration Formulas

Constant rule

∫ k dx = kx + C

A constant integrates to itself times x.

Power rule

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

For any n ≠ −1.

Reciprocal

∫ 1/x dx = ln|x| + C

The n = −1 exception.

Natural exponential

∫ eˣ dx = eˣ + C

Its own antiderivative.

General exponential

∫ aˣ dx = aˣ / ln a + C

Base a > 0, a ≠ 1.

Natural log

∫ ln x dx = x ln x − x + C

By parts.

Sine

∫ sin x dx = −cos x + C

Mind the minus sign.

Cosine

∫ cos x dx = sin x + C

No sign change.

Secant squared

∫ sec²x dx = tan x + C

Derivative of tangent.

Arctangent form

∫ 1/(1+x²) dx = arctan x + C

Inverse-trig result.

Arcsine form

∫ 1/√(1−x²) dx = arcsin x + C

Inverse-trig result.

Linearity

∫ [a·f + b·g] dx = a∫f + b∫g

Integrate term by term.

The Main Integration Techniques

Substitution

Reverse the chain rule

When an integrand contains a composite function together with (a constant multiple of) the derivative of its inner part, let u be the inner function. Then du absorbs the extra factor and the integral collapses to a standard form. Example: ∫2x·cos(x²) dx with u = x² becomes ∫cos u du = sin(x²) + C.

Integration by parts

Reverse the product rule

For a product of unlike functions, use ∫u dv = u·v − ∫v du. Choose u by the LIATE priority so that differentiating it eventually simplifies. Example: ∫x·eˣ dx with u = x, dv = eˣ dx gives x·eˣ − ∫eˣ dx = x·eˣ − eˣ + C.

Partial fractions

Split a rational function

A proper rational function whose denominator factors into distinct linear terms can be rewritten as a sum of simpler fractions, each of which integrates to a logarithm. Example: ∫1/(x²−4) dx splits into ¼ln|x−2| − ¼ln|x+2| + C.

Linearity

Integrate term by term

The integral of a sum is the sum of the integrals, and constant factors pull straight out. This is why any polynomial integrates instantly: you apply the power rule to each term and add the results, then attach a single + C at the end.

Common Integration Mistakes

  1. 1

    Forgetting the constant of integration

    Every indefinite integral needs + C. Without it you have found one antiderivative, not the family of all of them — and you cannot apply an initial condition later.

  2. 2

    Misapplying the power rule at n = −1

    ∫x⁻¹ dx is ln|x|, not x⁰/0. The power rule explicitly excludes n = −1 because it would divide by zero.

  3. 3

    Dropping the absolute value in logs

    ∫1/x dx = ln|x| + C. Writing ln(x) silently restricts the domain to positive x and loses half the solution.

  4. 4

    Integrating a product factor by factor

    ∫f·g dx is not ∫f dx · ∫g dx. Products require the product-in-reverse rule (integration by parts) or a substitution — never term-by-term multiplication.

  5. 5

    Chain rule confusion in reverse

    ∫cos(2x) dx = ½sin(2x) + C, not sin(2x). The inner derivative 2 must be divided out — a linear substitution the calculator handles automatically.

  6. 6

    Sign slip on sine and cosine

    ∫sin x dx = −cos x, while ∫cos x dx = +sin x. The minus sign lives with the sine integral; mixing them up is the classic trig-integral error.

Worked Examples

Power rule

∫ x² dx

Add one to the exponent and divide by the new exponent: x²⁺¹ / (2+1) = x³/3. Answer: x³/3 + C. Check: d/dx(x³/3) = 3x²/3 = x². ✓

Linearity

∫ (3x² + 2x + 1) dx

Integrate each term with the power rule: 3·x³/3 + 2·x²/2 + x = x³ + x² + x. Answer: x³ + x² + x + C. Check: differentiating returns 3x² + 2x + 1. ✓

Special forms

∫ sin x dx and ∫ eˣ dx

The sine integrates to −cos x + C (note the minus), and the natural exponential is its own antiderivative: ∫eˣ dx = eˣ + C. These two are worth memorising cold.

Reciprocal

∫ 1/x dx

This is the power rule's forbidden case n = −1, and it evaluates to the natural logarithm with an absolute value: ln|x| + C. Valid for both positive and negative x.

Methodology you can verify

Every integral is computed by a symbolic engine that parses your expression into an abstract syntax tree, applies the standard integration rules (power, exponential, logarithmic, trigonometric, substitution, integration by parts, and partial fractions), and then confirms the result by differentiating it and comparing against the original integrand at multiple sample points. When a function has no elementary antiderivative, the calculator states the limitation rather than returning a wrong answer. Read more on the methodology and editorial policy pages.

Frequently Asked Questions

An antiderivative of a function f(x) is a function F(x) whose derivative is f(x) — that is, F′(x) = f(x). It is the reverse of differentiation: instead of finding a rate of change, you reconstruct the original quantity from its rate. Because the derivative of any constant is zero, every function has infinitely many antiderivatives that differ only by an added constant, which is why the indefinite integral is written with a constant of integration, F(x) + C. For example, an antiderivative of 2x is x², and so is x² + 7 or x² − 100.

The words are closely related but not identical. An antiderivative (or indefinite integral) is a function F(x) + C whose derivative is the integrand; it produces a family of functions. A definite integral, written with limits like ∫ₐᵇ f(x) dx, is a single number — the signed area under the curve between a and b. The Fundamental Theorem of Calculus connects them: you evaluate a definite integral by finding an antiderivative F and computing F(b) − F(a). This calculator computes indefinite integrals (antiderivatives).

Because differentiation destroys constant information. If F(x) is an antiderivative of f(x), then F(x) + C has exactly the same derivative for any constant C, since the derivative of a constant is zero. So there is not one antiderivative but an entire family of parallel curves, all with the same slope at every point. The + C represents that whole family. You only pin down a single value of C when you are given an extra piece of information — an initial condition such as F(0) = 3 — that selects one specific curve.

It parses your expression into an abstract syntax tree with a built-in symbolic engine, then applies the standard integration rules: the constant and power rules, the reciprocal (natural log) rule, the exponential and general-exponential rules, logarithmic, trigonometric, inverse-trigonometric and hyperbolic rules, plus u-substitution, integration by parts, and partial fractions. It simplifies the result, appends the constant of integration, and then verifies the answer by differentiating it and comparing to the original integrand at several sample points. Finally it renders the worked solution, the rule used, the graph, and export options.

Yes. It knows the antiderivatives of all six trigonometric functions — sin, cos, tan, cot, sec, and csc — as well as the inverse-trigonometric functions arcsin, arccos, and arctan and the hyperbolic functions sinh, cosh, and tanh. It also handles common patterns such as sec²x → tan x, sin²x and cos²x via power-reduction, tan²x, products like sec x·tan x, and composites with a linear argument such as cos(2x) or sin(3x + 1) through substitution. Trigonometric integrals whose answer differentiates back to your input are marked as verified.

Yes. With the step-by-step option enabled (it is on by default), the calculator lists each stage of the solution: identifying the integral, applying the sum rule to split it into terms, integrating each term with the named rule and a short explanation of why it applies, adding the constant of integration, and finally verifying the result by differentiation. Every step shows the relevant expression so you can follow the reasoning and spot exactly where a hand-worked solution might have gone wrong.

You can enter expressions in ordinary calculator notation — use ^ for powers (x^2), * for multiplication (or simply write 2x and 3sin(x)), / for division, and function names like sin(x), cos(x), ln(x), sqrt(x), and e^x. Unicode symbols such as ², √, π, and × are accepted too. The calculator does not require full LaTeX input, but it outputs the answer as LaTeX (and MathML) that you can copy directly into a document, along with a plain-text and shareable-link version.

By differentiation. Because integration and differentiation are inverse operations, the derivative of a correct antiderivative must equal the original integrand. The calculator symbolically differentiates its own result and then compares it numerically against the function you entered at multiple sample points. If they agree everywhere the check runs, the answer is marked Verified. This is the most reliable self-check in calculus and guards against the tool ever silently returning a wrong answer.

Polynomials and constants; rational functions (through the f′/f logarithm rule, arctangent and arcsine forms, and simple partial fractions); exponential functions including eˣ and aˣ; natural and base-10 logarithms; all trigonometric, inverse-trigonometric, and hyperbolic functions; square roots and other powers; and many composite functions handled by linear substitution and integration by parts. When an expression has no antiderivative expressible in elementary functions — classic examples are eˣ², sin(x)/x, and 1/ln(x) — the calculator says so clearly rather than returning an incorrect result.

Yes. The symbolic engine applies exact integration rules, keeps rational coefficients as exact fractions rather than rounded decimals, and preserves symbolic constants like π and e. Every result is independently checked by differentiating it back to the original function, so an answer marked Verified is mathematically correct up to the arbitrary constant C. For the rare functions that lack an elementary antiderivative, the tool is transparent about the limitation and suggests a numerical (definite) integral instead.

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