integrate calculator

Definite Integrate Calculator

Compute ∫_a^b f(x) dx using numerical methods.

Function f(x)

Use x as the variable. Supported: + - * / ^, parentheses, sin, cos, tan, sqrt, log, exp, abs, pi, e.

Lower bound (a)

Upper bound (b)

Subintervals (n)

Larger n usually improves accuracy (but uses more computation).

Method

Quick examples:

What this integrate calculator does

This tool estimates a definite integral, meaning the signed area between a curve and the x-axis from a lower bound a to an upper bound b. Instead of requiring symbolic antiderivatives, it uses reliable numerical integration methods that work for many real-world functions.

If you have ever needed to estimate accumulated distance, total probability, total cost over time, or area under a signal curve, this is exactly the type of calculator you need.

How to use the calculator

1) Enter the function

Type your expression in terms of x. Use explicit multiplication (for example, 2*x, not 2x).

Polynomial: 3*x^3 - 2*x + 1
Trigonometric: sin(x), cos(x)^2
Exponential/log: exp(-x), log(x)

2) Set bounds and subintervals

Choose the lower and upper limits. Then set n, the number of subintervals. Higher values generally increase precision, especially for rapidly changing functions.

3) Choose a numerical method

Simpson's Rule: Often the most accurate of the three for smooth functions.
Trapezoidal Rule: Simple, stable, and widely used.
Midpoint Rule: Good performance for many curved functions.

Understanding the methods

Trapezoidal Rule

The interval is split into trapezoids; each area is summed. Think of replacing the curve with straight line segments between sample points.

Midpoint Rule

Each slice uses the function value at the midpoint. This often reduces error versus endpoint-based approximations on curved segments.

Simpson's Rule

Uses quadratic arcs across pairs of intervals. For smooth functions, this method usually converges quickly and can provide high accuracy with moderate n.

Practical tips for better results

Start with n = 100 or 200, then increase to check stability of the result.
If your function oscillates quickly, increase n substantially.
Watch domain restrictions (for example, log(x) requires x > 0).
For Simpson's Rule, an even n is required; this calculator auto-adjusts if needed.

Example use cases

Numerical integration is everywhere: physics (work and displacement), data science (area under curves), economics (accumulated value), and engineering (signal energy). A fast integrate calculator helps you move from equation to decision without getting stuck in manual computation.

FAQ

Can this compute indefinite integrals?

No. This page computes definite integrals numerically on a specific interval.

Why do two methods give slightly different answers?

Each method approximates curvature differently. As n increases, results usually converge toward the same value.

What if the result seems wrong?

Check syntax, domain validity, and increase subintervals. If the function has discontinuities, split the interval into valid pieces and integrate each piece separately.