Antiderivative Calculator
Compute indefinite integrals (antiderivatives) of polynomials, trigonometric, exponential, and logarithmic functions instantly. Each result includes the integration rule applied, step-by-step working, and the constant of integration +C.
How to Use This Antiderivative Calculator
Four simple steps to compute any antiderivative with full step-by-step working.
Choose Function Type
Select the tab for your function type — Polynomial, Trigonometric, Exponential, or Logarithmic. This filters the quick examples and hints shown.
Click an Example or Type Your Own
Click any quick-example button to load a preset function, or type your own in the input field using standard notation: ^ for powers, * for multiplication, and function names like sin, cos, exp, ln.
Set Variable & Options
The variable defaults to x but can be changed to t, u, or any single letter. Optionally set a multiplier to scale the function, and choose your constant name (C by default).
Read Your Result
See the antiderivative, the integration rule applied, full step-by-step working, and a differentiation check confirming the result is correct.
Standard Antiderivative Rules — Quick Reference
The integration rules used by this calculator — the same rules taught in every calculus course worldwide.
| f(x) | ∫ f(x) dx | Rule Name | Notes / Conditions |
|---|---|---|---|
| xⁿ | xⁿ⁺¹/(n+1) + C | Power Rule | n ≠ -1 |
| 1/x = x⁻¹ | ln|x| + C | Log Rule | Special case of power rule when n = -1 |
| k (constant) | kx + C | Constant Rule | k is any real number |
| sin(x) | -cos(x) + C | Trig Rule | Basic sine antiderivative |
| cos(x) | sin(x) + C | Trig Rule | Basic cosine antiderivative |
| tan(x) | -ln|cos(x)| + C | Trig / Log Rule | Derived from substitution |
| sec²(x) | tan(x) + C | Trig Rule | Antiderivative of sec squared |
| csc²(x) | -cot(x) + C | Trig Rule | Antiderivative of csc squared |
| eˣ | eˣ + C | Exponential Rule | e is Euler's number ≈ 2.71828 |
| aˣ | aˣ/ln(a) + C | Exponential Rule | a > 0, a ≠ 1 |
| ln(x) | x·ln(x) - x + C | Integration by Parts | x > 0 |
| 1/√(1-x²) | arcsin(x) + C | Inverse Trig Rule | |x| < 1 |
| 1/(1+x²) | arctan(x) + C | Inverse Trig Rule | All real x |
Understanding Antiderivatives and Indefinite Integrals
An antiderivative of a function f(x) is any function F(x) whose derivative equals f(x). In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x). The process of finding an antiderivative is called antidifferentiation or indefinite integration, and the result is written as ∫ f(x) dx = F(x) + C.
The constant of integration (+C) appears in every antiderivative because differentiation destroys constants — when you differentiate x² + 5, the 5 disappears. When you reverse the process, you cannot know what constant was there, so the answer must represent the entire family of functions that differ only by a constant.
The relationship between differentiation and integration is the Fundamental Theorem of Calculus — one of the most important results in all of mathematics. Part 1 states that if F(x) = ∫f(t)dt from a to x, then F'(x) = f(x). Part 2 provides the evaluation formula that allows definite integrals to be computed from antiderivatives.
- Power Rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
- Special case: ∫ x⁻¹ dx = ln|x| + C
- ∫ sin(x) dx = -cos(x) + C (note the negative sign)
- ∫ eˣ dx = eˣ + C (unique — equals its own antiderivative)
- Sum Rule: ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
- Constant Multiple: ∫ k·f(x) dx = k · ∫ f(x) dx
∫ Why -cos(x) for sin(x)?
The antiderivative of sin(x) is -cos(x) + C — the negative sign surprises many students. The reasoning: d/dx[-cos(x)] = -(-sin(x)) = sin(x). Verify it yourself: differentiate -cos(x) and you get sin(x). The antiderivative of cos(x), by contrast, is +sin(x) without the negative: d/dx[sin(x)] = cos(x). These signs are a constant source of errors on calculus exams — memorise them both carefully.
📑 The Power Rule — Why It Works
The Power Rule for integration is the reverse of the Power Rule for differentiation. Since d/dx[xⁿ] = nxⁿ⁻¹, to reverse this process you increase the exponent by 1 and divide by the new exponent: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C. The division by (n+1) compensates for what differentiation would multiply by. The only exception is n = -1, because you cannot divide by zero — this case gives ln|x| + C instead.
✅ Verifying Any Antiderivative
You can always verify an antiderivative by differentiating it — if you get back your original function, the antiderivative is correct. This is how our calculator's verification step works: it differentiates the computed result and checks that it matches the original input. If they match, the result is confirmed. This self-checking property of integration makes it one of the few areas of mathematics where you can easily verify your own answer.
📚 When Integration Requires Special Techniques
The basic rules handle many common functions, but more complex functions require advanced techniques. Integration by Parts handles products like x·sin(x) or x·eˣ using the formula ∫u dv = uv - ∫v du. Substitution (u-substitution) handles composite functions like ∫ sin(2x) dx by substituting u = 2x. Partial fractions handles rational functions by decomposing them into simpler fractions. Trigonometric substitution handles expressions with √(a²-x²).
Antiderivative Calculator FAQs
Common questions about antiderivatives, indefinite integrals, and how to use this calculator.
Explore More SpotDown Calculators
More free tools for math, education, finance, and everyday planning.
About This Antiderivative Calculator
This calculator applies standard integration rules — Power Rule, Constant Rule, Trigonometric rules, Exponential rules, and Logarithmic rules — to compute antiderivatives of common functions. Results include the integration rule applied, step-by-step working, and a differentiation-based verification check. The calculator handles sums and differences of terms using the Sum Rule of integration.
This tool is designed for educational purposes and handles standard calculus functions. Very complex expressions, products of functions requiring integration by parts, composite functions requiring substitution, and partial fractions may not be computable — use the function type tabs and examples for best results. For formal academic work, verify results manually using differentiation.
Ready to Compute an Antiderivative?
Use the free antiderivative calculator above or explore all our math and education tools.
∫ Compute Antiderivative All Math Tools