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Integral Calculator: Compute Antiderivatives and Definite Integrals with Plots
This tool evaluates indefinite integrals (antiderivatives) and definite integrals numerically and symbolically. Enter a function of x, set the bounds for definite integrals, and see the antiderivative, its graph, and the computed area under the curve.
Enter a function and bounds, then click Compute Integral to see results.
The antiderivative F(x) is the function whose derivative is f(x). For a definite integral from a to b, the result is F(b) – F(a), which equals the signed area under f(x) between those bounds. If the numerical and symbolic results differ, the function may not have a closed-form antiderivative; the numerical estimate remains valid. To verify symbolic results, consult a table of integrals or a computer algebra system.
What this calculator assumes: The input is a real-valued function of a single real variable x. It covers continuous functions on closed intervals and many standard discontinuous cases. It excludes functions with singularities within the interval unless the integral converges. The symbolic engine uses pattern matching against common integral forms; it does not guarantee a closed-form solution exists. Numerical integration uses the trapezoidal rule with the specified step count and is accurate to within typical floating-point precision for well-behaved functions.
When not to use: Do not rely on this for safety-critical, financial, or legal decisions. For improper integrals or highly oscillatory functions, increase the step count or use specialized software.
How the integral is computed
The calculator first attempts a symbolic antiderivative using a table of common integral forms. If successful, it evaluates F(b) – F(a) for definite integrals. Simultaneously, it computes a numerical estimate by integrating f(x) over [a, b] using the trapezoidal rule with n steps. The plot shows F(x) across the interval; for indefinite integrals, it uses an arbitrary constant C=0.
Worked examples bound to the live calculator
Compute ∫₀¹ x² dx. The antiderivative is x³/3. Evaluating from 0 to 1 gives 1/3 ≈ 0.3333.
Compute ∫₀^π sin(x) dx. The antiderivative is -cos(x). Evaluating from 0 to π gives -cos(π) – (-cos(0)) = 2.
Common pitfalls when integrating
Ignoring domain restrictions: Functions like 1/x or log(x) are undefined at x=0 or x≤0. If your interval includes such points, the integral may diverge or require a Cauchy principal value. The calculator will flag non-finite outputs; adjust bounds to avoid singularities.
Assuming every function has a closed-form antiderivative: Many elementary functions (e.g., e^(-x²), sin(x)/x) have no antiderivative expressible in elementary terms. The calculator will return a numerical result even when symbolic integration fails.
Miscounting signs in definite integrals: The definite integral from a to b is F(b) – F(a). Reversing the bounds changes the sign. If you swap a and b, the result should negate; verify this to catch sign errors.
Unusual boundary scenarios
Infinite bounds: For improper integrals like ∫₁^∞ 1/x² dx, enter Infinity as the upper bound. The calculator uses a large finite approximation (1e10) and checks convergence. The result should approach 1 for this example.
Zero-width interval: If a = b, the definite integral is zero by definition. The calculator handles this edge case explicitly.
Comparison: Symbolic vs. Numerical Results
| Scenario | Symbolic Result | Numerical Result | When to Use |
|---|---|---|---|
| Polynomial, trigonometric, exponential functions | Exact closed form | Approximate | Prefer symbolic for exact values; use numerical to verify. |
| Functions without elementary antiderivatives | Not available | Approximate | Numerical is the only option. |
| Discontinuous or piecewise functions | May fail | Approximate if continuous on subintervals | Split the interval at discontinuities. |
Frequently asked questions
Can this calculator handle double or triple integrals?
No. This tool computes single-variable integrals only. For multivariable calculus, use specialized software like SymPy, Mathematica, or Wolfram Alpha.
Why does the numerical result differ from the symbolic result?
For functions with closed-form antiderivatives, the difference is due to numerical approximation error (controlled by the step count). For functions without elementary antiderivatives, only the numerical result is meaningful.
How do I integrate a piecewise function?
Split the integral into subintervals where the function is defined by a single expression, compute each part separately, and sum the results. For example, ∫₋₁¹ |x| dx = ∫₋₁⁰ -x dx + ∫₀¹ x dx.
Sources and references
- Integral — Wikipedia: Overview of integration techniques and definitions.
- Trapezoidal rule — Wikipedia: Numerical integration method used for the numerical estimate.
- SymPy Integrals Tutorial (v1.12): Symbolic integration reference (methodology inspiration).