Configure the integral
Use y = arcsin(ax + b). The calculator checks real values automatically.
Formula used
For a nonzero coefficient, use the following standard identity.
For a = 0, the input is constant.
For definite integrals, calculate F(upper) − F(lower). Degree output multiplies angle-based results by 180 / π.
How to use this calculator
- Enter coefficient a and constant b for arcsin(ax + b).
- Select indefinite or definite calculation mode.
- Enter lower and upper bounds for a definite result.
- Choose radians or degrees and set display precision.
- Set a graph interval that overlaps the real domain.
- Press Calculate Integral and review the result above.
- Use CSV or PDF export to save the calculation.
Example data table
| Example | Input function | Bounds | Result in radians |
|---|---|---|---|
| Basic indefinite | arcsin(x) | Not required | x·arcsin(x) + √(1 − x²) + C |
| Bounded area | arcsin(x) | 0 to 1 | π / 2 − 1 ≈ 0.570796 |
| Shifted input | arcsin(2x − 1) | 0 to 1 | 0 |
| Constant input | arcsin(0.5) | 0 to 2 | π / 3 ≈ 1.047198 |
Understanding the integral of arcsin
Understanding the Integral of Arcsin
The inverse sine function returns an angle from a valid input. Its real input must stay between negative one and one. This calculator studies expressions shaped like arcsin(ax + b). It can find an antiderivative or evaluate a definite integral. It also checks the real domain before calculation.
Why the Formula Works
Integration by parts produces the main result. Let u equal arcsin(ax + b). Let dv equal dx. Differentiating u creates a square-root denominator. The remaining integral simplifies after substitution. The final expression contains both an arcsine term and a square-root term. Differentiating that result returns the original integrand.
Handling Coefficients
The coefficient a changes the antiderivative. When a is nonzero, divide the final formula by a. The constant b shifts the arcsine input horizontally. These values also decide where the expression remains real. The calculator reports the valid x interval automatically. A zero value for a needs separate handling. Then the integrand is constant because its input no longer changes.
Definite Integral Checks
A definite integral needs lower and upper bounds. Every input between those bounds must keep ax + b inside the real arcsine range. Because the inner expression is linear, checking both endpoints is enough. The calculator evaluates the antiderivative at each bound. It then subtracts the lower result from the upper result. Reversing the bounds changes the sign.
Radians and Degrees
Calculus formulas normally use radians. This calculator can also display angles in degrees. Degree mode multiplies the angle-based result by 180 divided by pi. Use radians for standard textbook work and derivative verification. Use degrees only when your application requires degree-valued inverse sine outputs. Keep the selected unit consistent while comparing results.
Graph and Verification Tools
The graph shows the integrand over your chosen interval. Invalid values are omitted from the real plot. The domain panel helps you choose suitable limits. The derivative check restates the function recovered from the antiderivative. Numerical formatting controls the displayed precision. CSV export saves a compact record. PDF export creates a printable summary. These tools make reviewing classwork, engineering inputs, and examples easier. Use the reset control to restore useful sample values. Small steps prevent domain mistakes. Save exported results with units, coefficients, bounds, and the displayed precision for future review and calculation records.
Frequently asked questions
1. What does this calculator integrate?
It integrates functions in the form arcsin(ax + b). It can return an indefinite antiderivative or a definite numerical result between two bounds.
2. Why must the arcsin input stay between −1 and 1?
Real-valued arcsin accepts only inputs from −1 through 1. Outside that range, the usual real inverse sine value does not exist.
3. What happens when a equals zero?
The input ax + b becomes the constant b. The integrand is then constant, so the antiderivative is x·arcsin(b) + C, provided b is valid.
4. Does the calculator support definite integrals?
Yes. Select definite mode, enter both bounds, and the calculator evaluates F(upper) − F(lower). It rejects bounds outside the real domain.
5. Are degree results suitable for standard calculus work?
Radians are normally expected in calculus. Degree mode is useful for applications needing degree-based inverse sine outputs. Keep units consistent during checks.
6. How is the domain found?
The calculator solves −1 ≤ ax + b ≤ 1. For nonzero a, this gives a closed x interval. For zero a, the domain is either all real numbers or none.
7. Why does the formula contain a square root?
Differentiating arcsin(u) produces 1 divided by √(1 − u²). Integration by parts leaves a related term, which becomes √(1 − u²) in the answer.
8. What does the optional evaluation point do?
For an indefinite calculation, it reports the integrand and antiderivative at one x value. The displayed antiderivative uses C = 0 for that point check.
9. Why is part of my graph missing?
The plot hides x values where ax + b falls outside −1 to 1. Adjust the graph range or coefficients to view valid real points.
10. Can I export the results?
Yes. After a successful calculation, use Download CSV for a data file or Download PDF for a printable result summary.
11. How can I verify the answer manually?
Differentiate the antiderivative. The result should equal arcsin(ax + b). For a definite integral, substitute each bound into F(x) and subtract.