Integral Calculator
Calculate definite integrals using numerical methods like Simpson's Rule, Trapezoidal Rule, and more
Integral Setup
Supported: +, -, *, /, ^, sin, cos, tan, ln, log, sqrt, pi, e
High accuracy for smooth functions
Higher values = more accurate but slower calculation
How to Use the Integral Calculator
Enter Mathematical Function
Input your function using standard mathematical notation. Supported functions include polynomials, trigonometric functions (sin, cos, tan), exponential (e^x), logarithmic (ln, log), and square root (sqrt).
Set Integration Limits
For definite integrals, enter the lower limit (a) and upper limit (b). The calculator will evaluate the integral from a to b. Make sure the upper limit is greater than the lower limit.
Choose Integration Method
Select from Simpson's Rule (most accurate), Trapezoidal Rule (reliable), Midpoint Rule (good for continuous functions), or Romberg Integration (very high precision). Each method has different accuracy and computational requirements.
Adjust Precision
Set the number of intervals for numerical integration. Higher values provide more accurate results but take longer to calculate. For most functions, 1000 intervals provide good accuracy.
Review Results
The calculator shows the integral value, method used, integration interval, average value of the function, and estimated error. Use the visualization to understand the function behavior.
Integration Tips
Use Simpson's Rule for smooth functions as it provides the highest accuracy among basic methods
For functions with discontinuities, try the Midpoint Rule which handles them better than other methods
Increase precision (intervals) for more accurate results, especially for rapidly changing functions
Check that your function syntax is correct - use * for multiplication (e.g., 2*x, not 2x)
Use parentheses to ensure correct order of operations, especially with complex expressions
For trigonometric functions, make sure your limits make sense (e.g., avoid discontinuities)
The Romberg method automatically adjusts precision and often provides the most accurate results
Verify results by comparing different methods - they should give similar answers for well-behaved functions
For very large or small numbers, the calculator automatically handles scientific notation
Use pi for π and e for Euler's number in your expressions