Calculator Inputs
Function Graph
The graph shows the selected integrand multiplied by k. Undefined points are skipped.
Formula Used
Definite integral rule:
∫ from p to q kf(x) dx = k[F(q) − F(p)]
| Form | Substitution | Antiderivative | Domain |
|---|---|---|---|
| ∫ dx / √(a² − x²) | x = a sin θ | sin⁻¹(x / a) + C | |x| < a for the integrand. Endpoints may be used for some definite limits. |
| ∫ √(a² − x²) dx | x = a sin θ | (x / 2)√(a² − x²) + (a² / 2)sin⁻¹(x / a) + C | |x| ≤ a |
| ∫ dx / (a² + x²) | x = a tan θ | (1 / a)tan⁻¹(x / a) + C | All real x |
| ∫ dx / √(a² + x²) | x = a tan θ | asinh(x / a) + C | All real x |
| ∫ √(a² + x²) dx | x = a tan θ | (x / 2)√(a² + x²) + (a² / 2)asinh(x / a) + C | All real x |
| ∫ dx / √(x² − a²) | x = a sec θ | ln|x + √(x² − a²)| + C | |x| > a |
| ∫ √(x² − a²) dx | x = a sec θ | (x / 2)√(x² − a²) − (a² / 2)ln|x + √(x² − a²)| + C | |x| ≥ a |
| ∫ dx / (x² − a²) | x = a sec θ or partial fractions | (1 / 2a)ln|(x − a) / (x + a)| + C | x ≠ ±a |
| ∫ x dx / √(a² − x²) | u = a² − x² | −√(a² − x²) + C | |x| < a |
| ∫ x dx / √(a² + x²) | u = a² + x² | √(a² + x²) + C | All real x |
How to Use This Calculator
- Select the integral pattern that matches your expression.
- Enter the positive value of
a. - Add a multiplier if your integral has a constant outside.
- Enter lower and upper limits for a definite answer.
- Enter an x value to test the integrand at one point.
- Press calculate and review the result above the form.
- Use the graph to inspect shape and domain restrictions.
- Download the answer as CSV or PDF for records.
Example Data Table
| Example | a | Limits | Substitution | Antiderivative |
|---|---|---|---|---|
| ∫ dx / √(25 − x²) | 5 | 0 to 3 | x = 5 sin θ | sin⁻¹(x / 5) + C |
| ∫ dx / (9 + x²) | 3 | 0 to 4 | x = 3 tan θ | (1 / 3)tan⁻¹(x / 3) + C |
| ∫ √(x² − 16) dx | 4 | 5 to 8 | x = 4 sec θ | (x / 2)√(x² − 16) − 8ln|x + √(x² − 16)| + C |
Understanding Trig Substitution Integrals
Why This Method Matters
Trig substitution is a powerful integration method. It changes difficult radical expressions into simpler trigonometric forms.
The method is common in calculus, physics, geometry, and engineering. It works best when the expression contains
a² − x², a² + x², or x² − a². These patterns match the main Pythagorean identities.
Once the expression is transformed, the radical often cancels or becomes easier to integrate.
Choosing the Right Substitution
Use x = a sin θ for a² − x². This choice creates a² cos² θ.
Use x = a tan θ for a² + x². This creates a² sec² θ.
Use x = a sec θ for x² − a². This creates a² tan² θ.
These choices are not random. They are based on identities that remove square roots.
Definite Integral Handling
This calculator uses the antiderivative rule for definite integrals. It finds F(upper) − F(lower).
Then it multiplies the result by the constant k. Domain checks are important.
Some forms are undefined outside allowed intervals. Some forms have vertical asymptotes.
The calculator warns you when limits cross restricted regions.
Graph and Export Benefits
The graph helps you see shape, endpoints, gaps, and steep changes. This is useful before trusting a numerical result. The CSV export is useful for spreadsheets. The PDF export is useful for homework notes, reports, or saved examples. The formula table also makes the calculator a study guide. You can compare forms and review substitutions quickly.
FAQs
1. What is trig substitution?
Trig substitution is an integration method that replaces x with a trigonometric expression. It simplifies radicals like a² − x², a² + x², and x² − a².
2. Which substitution should I choose?
Use x = a sin θ for a² − x². Use x = a tan θ for a² + x². Use x = a sec θ for x² − a².
3. Can this calculator solve every integral?
No. It handles common trig substitution forms. Complex expressions may need algebraic simplification, completing the square, or another integration method first.
4. Why does the calculator show domain warnings?
Some radicals and denominators are undefined for certain x values. A warning means the selected limits cross an invalid region or singular point.
5. What does multiplier k mean?
The multiplier k is a constant placed outside the integral. The final antiderivative and definite result are multiplied by this value.
6. Why is the graph sometimes broken?
A broken graph usually means the function is undefined over part of the range. The calculator skips those invalid points.
7. Does the calculator give exact symbolic answers?
It gives formula-based antiderivatives and numerical definite results. Exact symbolic simplification beyond the listed patterns is not attempted.
8. Can I export my result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean printable summary of the calculated result.