What an Integral Represents
Integration is the calculus operation that accumulates quantities. In geometry, a definite integral can represent net area under a curve. In physics, integrals accumulate rates (like velocity) into totals (like distance). In statistics, integrals appear in probability density functions and expectations. In finance, integration helps model continuous change and discounting.
This Integral Calculator supports two common tasks: finding an antiderivative (indefinite integral) and computing a numeric value over an interval (definite integral). When a closed-form antiderivative is recognized, the tool shows rule-based steps. When not, it uses Simpson’s rule to approximate the definite integral reliably.
Indefinite Integrals and the Constant of Integration
An indefinite integral returns a family of functions because derivatives lose constants. That is why you often add + C. If F′(x)=f(x), then ∫ f(x) dx = F(x) + C. In this tool you can toggle whether to append + C to the final expression.
Definite Integrals, Net Area, and Area Under the Curve
A definite integral ∫ab f(x) dx computes net signed area: parts of the curve above the x-axis contribute positive area, while parts below contribute negative area. Sometimes you need physical area, which is ∫ |f(x)| dx. The “Compute Area” option lets you choose net signed area or absolute area under the curve.
favg = (1 / (b − a)) · ∫ab f(x) dx
How the Calculator Chooses a Method
In Auto mode, the calculator tries to compute a symbolic antiderivative for common patterns (polynomials, sums, constant multiples, exp, ln, sin, cos, simple power chains). If it cannot confidently produce a symbolic form, it switches to numeric integration for the definite integral.
If you already know the integral is not elementary or the expression is complex, choose Numeric mode. If you need an exact symbolic result and your expression is a standard form, choose Symbolic only.
Simpson’s Rule for Numeric Definite Integrals
Simpson’s rule is a standard numerical method that approximates ∫ f(x) dx by fitting quadratic arcs across small subintervals. It tends to be very accurate for smooth functions when you use a sufficiently large even number of subintervals.
Oscillatory or rapidly changing functions may require a larger N. The calculator accepts an even N and will adjust if you enter an odd value.
Integration Tables and CSV Export
The f(x) Table tab generates a list of x values and function values over an interval. This helps you:
- Plot the function in a spreadsheet
- Inspect discontinuities or rapid changes
- Validate numeric integration bounds
- Share computed values with others
Use CSV export to move the table into Excel or Google Sheets.
Limitations and Practical Tips
Symbolic integration is harder than differentiation. Many valid integrals do not have elementary closed forms, and others require advanced techniques (integration by parts, partial fractions, special functions). This tool provides dependable results by combining basic symbolic rules with numeric integration for definite bounds.
- Use explicit multiplication: 2*x not 2x
- Use exp(x) for e^x
- Use ln(x) for natural log
- Use parentheses to avoid ambiguity: sin(x)/x, not sin x / x
FAQ
Integral Calculator – Frequently Asked Questions
Answers about indefinite and definite integrals, methods, and interpretation.
An integral is a calculus operation that accumulates quantities. Indefinite integrals represent families of antiderivatives, while definite integrals compute net area under a curve between two bounds.
Indefinite integrals produce an antiderivative expression plus a constant of integration (C). Definite integrals evaluate the antiderivative at bounds to produce a numeric result.
Yes. It shows rule-based steps for supported forms (power rule, exponentials, basic trig, simple substitutions). For more complex expressions, it uses numeric integration for definite bounds and provides method notes.
For definite integrals, the calculator can approximate results using Simpson’s rule with many subintervals. This is useful when a closed-form antiderivative is difficult or not supported.
Area under the curve is the integral of |f(x)| over an interval. The standard definite integral gives net signed area, which can be negative where the function is below the x-axis.
The average value on [a,b] is (1/(b−a))·∫[a,b] f(x) dx. This calculator computes it for definite integrals.
Yes for common forms such as sin(x), cos(x), exp(x), ln(x), and simple composites. Complex combinations may require numeric definite integration.
Yes. You can generate a table of x and f(x) across an interval and export it to CSV for plotting or analysis.
Indefinite integral results are exact when a supported symbolic form is recognized. Definite integrals may be exact (via antiderivative) or approximate (via numeric integration) depending on the expression.