sin(x), cos(x), exp(x), ln(x), sqrt(x). Constants: pi, e.What this integral calculator does
This tool computes definite integrals numerically. In plain language, it estimates the accumulated area under a curve between two x-values. Instead of trying to find a symbolic antiderivative by hand, the calculator samples the function across many small slices and adds them together.
It is ideal for quick checks, homework verification, engineering approximations, and exploring functions where a closed-form antiderivative is difficult or impossible to write down.
How to enter functions correctly
Basic syntax
Enter expressions in terms of x. Use operators like +, -, *, /, and ^.
Parentheses are strongly recommended for clarity.
x^3 - 2*x + 5sin(x)sqrt(1 + x^2)ln(x)(same as natural log)exp(-x^2)
Constants and functions supported
The parser supports common math functions including trigonometric and logarithmic forms:
sin, cos, tan, asin, acos, atan,
sqrt, abs, exp, log/ln, floor, ceil, and more.
Constants pi and e are also recognized.
Numerical methods available
Simpson's Rule
Simpson's Rule uses parabolic arcs to approximate the function over each pair of intervals. It often provides excellent accuracy for smooth functions and is generally the best default choice.
Trapezoidal Rule
This method connects points on the curve with straight segments, creating trapezoids. It is simple, reliable, and useful when you want transparent, stable behavior across many function types.
Midpoint Rule
Midpoint evaluates the function at the center of each subinterval. It can outperform trapezoidal estimates in some cases, especially when the function is reasonably smooth.
Tips for better accuracy
- Increase subintervals (n) for tighter approximations.
- Use Simpson's Rule for smooth functions whenever possible.
- Watch for singularities (like division by zero) inside your interval.
- For oscillating functions, use a higher n to capture behavior.
Common mistakes to avoid
A frequent issue is forgetting multiplication symbols. For example, write 2*x instead of 2x.
Also ensure your function is defined on the entire interval. If a point gives an undefined value (for example, ln(x) at
non-positive x), the calculator will report an error.
Why integrals matter
Integrals appear everywhere: computing distance from velocity, total cost from marginal cost, probability from density functions, and accumulated change in finance, biology, and physics. Understanding numerical integration gives you a practical way to solve real problems even when symbolic calculus gets complicated.