Integral Calculator
Find antiderivatives and evaluate definite integrals.
Try an example
Rules Used
Power Rule
∫xⁿ dx = xⁿ⁺¹/(n+1)
Constant Multiple Rule
∫c·f(x) dx = c·∫f(x) dx
Sum Rule
∫(f + g) dx = ∫f dx + ∫g dx
Step-by-Step Solution
Step 1 — Split the integral
∫x² dx + ∫3x dx + ∫2 dx
Step 2 — Integrate each term
∫x² dx = x³/3
∫3x dx = 3x²/2
∫2 dx = 2x
Step 3 — Combine
x³/3 + 3x²/2 + 2x + C
Evaluate Function
Enter an x-value to evaluate f(x).
Graph
Relation to Derivatives
Integration reverses differentiation.
If F(x) = x³/3 + 3x²/2 + 2x, then F'(x) = x² + 3x + 2.
Therefore ∫(x² + 3x + 2) dx = x³/3 + 3x²/2 + 2x + C.
This integral calculator finds the antiderivative (indefinite integral) of a function, or evaluates a definite integral over a specific interval - the reverse operation of differentiation, and the tool used to find quantities like total area under a curve.
Enter a function (and bounds, for a definite integral), and the calculator returns the result.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Indefinite integral of 3x² | ∫3x² dx = x³ + C | Power rule for integration: raise the exponent by one and divide by the new exponent. ∫3x² dx = 3 · (x³/3) + C = x³ + C. |
| Definite integral of (2x + 1) from 0 to 2 | 6 | Antiderivative = x² + x. Evaluate at bounds: (2² + 2) − (0² + 0) = 6 − 0 = 6. |
Indefinite vs. Definite Integrals
An indefinite integral returns a general antiderivative function, plus a constant of integration (+C), since many functions share the same derivative. A definite integral evaluates that antiderivative between two specific bounds, producing a single numeric value - commonly interpreted as the net area under the curve between those bounds.
Integration Reverses Differentiation
Integration and differentiation are inverse operations: integrating a function's derivative returns the original function (up to a constant). This relationship, formalized by the Fundamental Theorem of Calculus, is why integral tables often mirror derivative rules in reverse - the power rule for integration adds one to the exponent and divides by the new exponent, the opposite of differentiation's power rule.
How to Use the Integral Calculator
- Enter a function of x.
- Choose Indefinite or Definite Integral.
- If using a definite integral, enter the lower and upper bounds.
- Review the result.
- Open Step-by-Step Solution to understand how the integral was evaluated.
- Use the graph to visualize the function and the integrated interval.
Frequently Asked Questions
What is the difference between a definite and indefinite integral?
An indefinite integral returns a general antiderivative function plus a constant (+C). A definite integral evaluates that antiderivative between two bounds, giving a single numeric value representing net area under the curve.
How do I find the antiderivative of a function?
Apply integration rules in reverse of differentiation - for example, the power rule for integration turns x^n into x^(n+1)/(n+1), then add a constant of integration for an indefinite integral.
What does a definite integral represent geometrically?
It represents the net signed area between the function's curve and the x-axis over the given interval - area below the x-axis counts as negative.
How do I use this integral calculator?
Enter the known values, review the units or settings, and the calculator updates the result instantly. The formula and example on this page show how the answer is produced.
What does the Integral Calculator calculate?
Solve Integral problems online with fast math inputs and results. It is designed for fast browser-based calculations without sign-up, downloads, or manual spreadsheet setup.
What formula does this integral use?
The formula is: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (Fundamental Theorem of Calculus links integration and differentiation). The Fundamental Theorem of Calculus, proven in its modern form in the late 17th century as part of Newton and Leibniz's independent work, is considered one of the most important results in mathematics because it revealed that integration (finding area under a curve) and differentiation (finding rates of change) are inverse operations - unifying what had previously been two separate branches of mathematical problem-solving into a single coherent framework.
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Last updated: September 27, 2026.