What kind of integrals does this calculator compute?

This calculator computes numerical approximations of definite integrals of real-valued functions over finite intervals, that is ∫_a^b f(x) dx with real numbers a and b. It does not attempt to produce symbolic antiderivatives; instead it uses Simpson’s rule with a refinement step and an error estimate.

Is this a symbolic integral calculator like a CAS?

No. The tool is based on numerical quadrature rather than symbolic manipulation. It is ideal for building intuition, checking hand calculations, integrating functions that do not have simple closed-form antiderivatives, and working with empirical formulas. For exact symbolic integrals you should still use a computer algebra system or perform an analytic calculation.

How accurate are the numerical results?

For smooth functions on a finite interval, composite Simpson’s rule with a reasonable number of subintervals is generally very accurate. The calculator computes the integral with a base number of subintervals and again with twice as many, then uses the difference as a simple error estimate. Highly oscillatory functions, sharp spikes, or singularities inside the interval can reduce accuracy, in which case the warnings field and sample points help you diagnose issues.

Can I use this calculator for improper integrals or infinite limits of integration?

The current version focuses on proper integrals with finite endpoints. Some improper integrals can be approximated by choosing a large but finite bound and inspecting whether the value stabilises, but this requires care and mathematical judgement. For rigorous work with improper integrals it is better to use analytic techniques or specialised numerical routines.

Which functions and syntax are supported?

Use x as the variable and the operators +, -, *, / and ^ for powers, together with sin, cos, tan, asin, acos, atan, ln, log, sqrt, abs, exp, floor, ceil, max, min and the constants pi and e. For example: sin(x)/x, (1+x^2)*exp(-x^2), or max(0, 1-x). Trigonometric functions use radians, consistent with most programming languages and scientific software.

Integral Calculator (Numerical)

Definite integrals over [a, b]

Compute the definite integral \(\displaystyle \int_a^b f(x)\,dx\) of a real-valued function using composite Simpson’s rule with refinement and an error estimate. Ideal for engineering, data analysis, and teaching calculus.

Focus: high-quality numerical integration on a finite interval. For symbolic antiderivatives and step-by-step algebra, use a CAS – then check or explore the result here.

1. Define the integral

Function f(x)

Use x as the variable. Allowed: +, -, *, /, ^, sin, cos, tan, asin, acos, atan, ln, log, sqrt, abs, exp, floor, ceil, max, min, constants pi, e. Trigonometric functions use radians.

Integration interval

Lower limit a

Upper limit b

Works best when f(x) is continuous or reasonably smooth on [a, b].

Subintervals (for Simpson)

Must be even. The calculator will adjust it if necessary.

Decimals

Display precision for integral and sample values.

Sample points for table

Number of (x, f(x)) pairs shown in the sample table.

Quick examples:

2. Numerical results

∫_a^b f(x) dx (Simpson, refined)

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Absolute area ∫_a^b |f(x)| dx

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Average value on [a, b]

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Estimated error |I_2n − I_n|

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Function evaluations (approx.)

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Convergence note

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Warnings

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3. Sample points (x, f(x)) on [a, b]

Use this table to inspect how the function behaves on the integration interval and to spot peaks, sign changes or potential numerical issues.

x	f(x)

What is a definite integral?

The definite integral of a function \(f(x)\) over an interval \([a, b]\) is written as

\[ \int_a^b f(x)\,\mathrm{d}x. \]

Geometrically, for functions that stay above the x-axis, this represents the signed area under the graph of \(f(x)\) from x = a to x = b. When the function crosses the axis, areas above and below the axis can cancel.

In calculus, definite integrals are defined via limits of Riemann sums, and they form one half of the fundamental theorem of calculus, connecting antiderivatives and area.

Numerical integration with Simpson’s rule

Not every integral has a closed-form antiderivative. Even when it does, in practical engineering and data analysis it is often easier to evaluate the integral numerically. Composite Simpson’s rule approximates the function by parabolas over small subintervals, then sums the areas:

Divide \([a, b]\) into \(n\) subintervals (with \(n\) even) of width \(h = \dfrac{b-a}{n}\). Let \(x_k = a + kh\) and \(f_k = f(x_k)\). Then:

\[ \int_a^b f(x)\,dx \approx \frac{h}{3} \left( f_0 + f_n + 4 \sum_{k=1,3,\dots,n-1} f_k + 2 \sum_{k=2,4,\dots,n-2} f_k \right). \]

Our calculator computes this approximation with a base number of subintervals \(n\) and again with \(2n\) subintervals; the difference \(|I_{2n} - I_n|\) is used as a simple error estimate.

Signed area, absolute area and average value

The signed integral may be small even when the graph of \(f(x)\) has large positive and negative lobes that cancel. For that reason the tool also reports:

The absolute area \(\int_a^b |f(x)|\,dx\), which treats negative contributions as positive;
The average value of the function on \([a,b]\): \[ f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx. \]

These summaries are very useful in physics, signal processing and statistics (for example when computing mean energy or average load over time).

FAQ: integrals and this calculator

What is the difference between a definite and an indefinite integral?

An indefinite integral \(\int f(x)\,dx\) represents the family of antiderivatives of \(f\), written as \(F(x) + C\). A definite integral \(\int_a^b f(x)\,dx\) is a specific number, obtained (when an antiderivative exists) as \(F(b) - F(a)\). This calculator focuses on definite integrals, computed numerically.

Why do I sometimes get “NaN” or warnings?

Numerical integration assumes that the function is well-behaved on \([a, b]\). If your expression leads to division by zero, square roots of negative numbers, logarithms of non-positive values, or very steep spikes, some points may be undefined or extremely large. The calculator flags these situations so that you can adjust the interval, reformulate the problem, or use analytic methods.

How can I improve accuracy?

Try increasing the number of subintervals, as long as it stays within a reasonable range (e.g. up to a few thousands for smooth functions). You can also break the interval into pieces and integrate separately if the function behaves differently in different regions. Always interpret numerical results together with domain knowledge and, when important decisions are involved, cross-check with alternative methods.