Improper Integral Calculator
Numerically approximate improper integrals with infinite limits using adaptive truncation.
About Improper Integral Calculator
Evaluates improper integrals (those with infinite limits) by adaptively finding effective bounds where |f(x)| becomes negligible, then applying Simpson's rule on the truncated interval. Use ±9999 to represent ±∞. The calculator reports whether convergence was achieved.
Formula
Improper integral
∫[a,∞) f(x)dx ≈ ∫[a,T] f(x)dx where f(T) ≈ 0
- a
- Lower bound
- T
- Effective upper truncation point
Worked example
With the default values below, Improper Integral Calculator returns Integral value of 0.88622693. Change any input above and every figure here is recalculated in your browser.
Inputs used
- f(x)
- exp(-x^2)
- Lower bound (use -9999 for -∞)
- 0
- Upper bound (use 9999 for +∞)
- 9,999
- Subintervalsaccepts 100–100,000
- 2,000
Results
- Integral value
- 0.88622693
- Status
- Approximate (tail may contribute)
Step by step
- 1Values used
- 2Improper integral
- 3Integral value= 0.88622693
- 4Status= Approximate (tail may contribute)
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
How do I enter infinite limits?
Use -9999 for negative infinity and 9999 for positive infinity. The calculator adaptively determines where the function becomes negligible.
What if the integral diverges?
If f(x) does not decay fast enough, the calculator will report an approximate result with a warning that the tail may contribute.