find definite integral calculator - Aaron Graves, PhDude Replica

Definite Integral Calculator

Use this tool to approximate ∫_a^b f(x) dx with Simpson’s Rule.

Function f(x)

Use variable x, operators + - * / ^, and functions like sin, cos, ln, log, sqrt, abs, exp. Constants: pi, e.

Lower bound (a)

Upper bound (b)

Subintervals n (even number)

How to find a definite integral

A definite integral measures the accumulated quantity of a function over an interval. Geometrically, it is the signed area between the curve f(x) and the x-axis from a to b. In notation:

∫_a^b f(x) dx

Some integrals can be solved exactly with antiderivatives, while others are best handled numerically. This page gives you a practical way to compute accurate approximations quickly.

How this calculator works

1) Parse your function

The calculator reads your expression (for example x^3 - 2*x + 1), interprets standard math functions, and evaluates it at many points in your interval.

2) Apply composite Simpson’s Rule

Simpson’s Rule fits parabolas across subintervals to estimate area. For smooth functions, it is usually much more accurate than basic rectangle methods. The calculator computes the integral with n slices and also with 2n slices, then reports an error estimate from the difference.

3) Return a stable numeric result

The displayed value is the higher-resolution estimate. If your function has discontinuities or undefined points inside the interval, the calculator will return a warning.

Input guide and supported syntax

Variable: x
Operators: +, -, *, /, ^
Parentheses: use freely for grouping
Functions: sin, cos, tan, asin, acos, atan, sqrt, abs, ln, log, exp, floor, ceil, round
Constants: pi, e

Note: ln(x) means natural log and log(x) means base-10 log.

Examples you can try

x^2 from 0 to 2 → exact answer is 8/3
sin(x) from 0 to pi → exact answer is 2
1/(1+x^2) from 0 to 1 → exact answer is pi/4
exp(-x^2) from 0 to 1 → no elementary antiderivative (numerical method is ideal)

Common mistakes when finding definite integrals

Forgetting parentheses in expressions like 1/(x+1)
Using odd n with Simpson’s Rule (this calculator auto-corrects to even)
Integrating across a discontinuity without splitting the interval
Expecting exact symbolic output for functions that require numerical approximation

Final tip

For homework or engineering checks, combine both approaches: first understand the integral conceptually (area, accumulation, units), then use a numerical calculator to get a fast, reliable value. If the estimate changes little as n increases, your answer is typically stable.