Calculator
Use explicit multiplication such as 2*x. Supported functions include sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, log, log10, sqrt, abs, pi, and e.
Example data table
| Function | Interval type | Bounds | Known value |
|---|---|---|---|
| exp(-x) | Upper infinite | [0, ∞) | 1 |
| 1/(x^2) | Upper infinite | [1, ∞) | 1 |
| exp(-x^2) | Two-sided infinite | (-∞, ∞) | √π |
| 1/sqrt(x) | Left singular | [0, 1] | 2 |
| 1/sqrt(1-x) | Right singular | [0, 1] | 2 |
| 1/sqrt(x*(1-x)) | Both endpoint singularities | [0, 1] | π |
Formula used
Improper integrals are evaluated through limits or transformations that move difficult endpoints into a finite interval. The calculator then applies adaptive Simpson integration to the transformed problem.
- Upper infinite case: ∫[a,∞) f(x)dx = ∫[0,1] f(a + t/(1-t)) / (1-t)^2 dt
- Lower infinite case: ∫(-∞,b] f(x)dx = ∫[0,1] f(b - t/(1-t)) / (1-t)^2 dt
- Two-sided infinite case: ∫(-∞,∞) f(x)dx = ∫[0,1] f(tan(π(t-1/2))) π sec²(π(t-1/2)) dt
- Left singularity: x = a + (b-a)t², so dx = 2(b-a)t dt
- Right singularity: x = b - (b-a)t², so dx = 2(b-a)t dt
- Both endpoints singular: x = a + (b-a)(1-cos(πt))/2
The adaptive Simpson rule compares one large estimate with two half-interval estimates. When the difference becomes small enough, the interval is accepted and the integral estimate is refined.
How to use this calculator
- Enter a function of x using explicit multiplication.
- Choose the improper mode that matches your bounds or singular endpoint.
- Enter the lower and upper bound values when the mode needs them.
- Set tolerance, recursion depth, trim, and sample points.
- Press the evaluate button to view the result above the form.
- Review the error estimate, secondary estimate, and assessment line.
- Inspect the Plotly graph to see numerical behavior.
- Use the export buttons to save a CSV or PDF summary.
Improper integral evaluation guide
Why these integrals are different
An improper integral does not fit the standard finite-interval picture. The interval may stretch to infinity, or the integrand may grow without bound near an endpoint. In each case, the integral is defined by a limit. That limit may converge to a finite number, or it may diverge. A reliable calculator must handle both the mathematics and the numerical behavior carefully.
How the calculator works
This tool converts common improper cases into a finite interval. That step is important because finite intervals are easier to integrate numerically. After transformation, the calculator uses adaptive Simpson integration. The algorithm estimates an area on a large interval, splits the interval, and compares the smaller estimates with the larger one. If the difference falls below the requested tolerance, the local estimate is accepted. If not, the interval is split again.
How to interpret the result
The approximate integral is the main answer. The estimated error gives a practical sense of numerical uncertainty for the transformed integral. The secondary estimate is a stability check created by tightening the settings. When the primary and secondary values agree closely, the answer is usually dependable. If the stability note warns about sensitivity, the integral may still converge mathematically, but the chosen numeric settings or expression form may need adjustment.
When to change the settings
Use a smaller tolerance when you need more precision. Increase recursion depth when the integrand changes rapidly. Reduce the endpoint trim when the result depends strongly on the improper boundary and the transformed integrand remains stable. Increase sample points when you want a smoother graph. For classroom work, compare the output with known values from the example table to confirm that the entered function and mode are correct.
FAQs
1. What makes an integral improper?
An integral is improper when the interval is infinite or the integrand becomes unbounded at a limit point. Its value is defined through a convergent limit.
2. Can this calculator prove convergence?
No. It gives a numerical assessment based on transformed estimates, error size, and stability checks. Formal convergence proofs still depend on mathematical analysis.
3. Which functions can I enter?
You can enter expressions in x with standard operators and supported functions such as exp, log, sqrt, trig functions, absolute value, pi, and e.
4. Why do I need explicit multiplication?
The parser expects forms like 2*x and x*(x+1). Writing 2x or xsin(x) can be ambiguous for a compact calculator parser.
5. What does endpoint trim do?
Trim keeps the numerical routine away from the exact transformed endpoints. That avoids undefined floating-point evaluations while still capturing the limit behavior.
6. Why does the secondary estimate matter?
It acts as a quick stability check. Close agreement suggests a stable setup, while a large gap signals sensitivity near the improper boundary.
7. Why can a convergent integral still look unstable numerically?
Sharp tails, steep singular behavior, or an expression with large cancellation can challenge floating-point arithmetic. Better settings or algebraic simplification often help.
8. What should I do if the result seems wrong?
Check the mode, bounds, expression syntax, and tolerance. Then compare with a known example and inspect the graph for rapid growth or oscillation.