Improper Integral Calculator
Numerically approximate improper integrals with infinite limits using adaptive truncation.
Inputs
Enter integrand using x.
Integral value
0.88622693
Status
Approximate (tail may contribute)
Step by step
Values used
f(x) = exp(-x^2); Lower bound (use -9999 for -∞) = 0; Upper bound (use 9999 for +∞) = 9,999; Subintervals = 2,000
Improper integral
∫[a,∞) f(x)dx ≈ ∫[a,T] f(x)dx where f(T) ≈ 0
Integral value
= 0.88622693
Status
= Approximate (tail may contribute)
How it works
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
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.