These examples show sample antiderivative outputs for common settings.
| Function | k | a | b | Sample x | Approx. F(x) |
|---|---|---|---|---|---|
| arcsin(x) | 1 | 1 | 0 | 0.5 | 1.127825 |
| arccos(x) | 1 | 1 | 0 | 0.25 | -0.638717 |
| arctan(x) | 1 | 1 | 0 | 1 | 0.438825 |
| arcsec(x) | 1 | 1 | 0 | 2 | 0.777437 |
This calculator evaluates integrals of the form ∫ k · f-1(ax + b) dx. It uses substitution and integration by parts.
Let u = ax + b. Then dx = du / a, so ∫ k · f-1(ax + b) dx = (k / a) ∫ f-1(u) du + C.
| Inverse function | Base antiderivative with variable u | Domain reminder |
|---|---|---|
| sin-1(u) | u sin-1(u) + √(1 - u2) + C | |u| ≤ 1 |
| cos-1(u) | u cos-1(u) - √(1 - u2) + C | |u| ≤ 1 |
| tan-1(u) | u tan-1(u) - ½ ln(1 + u2) + C | All real u |
| cot-1(u) | u cot-1(u) + ½ ln(1 + u2) + C | All real u |
| sec-1(u) | u sec-1(u) - ln(|u| + √(u2 - 1)) + C | |u| ≥ 1 |
| csc-1(u) | u csc-1(u) + ln(|u| + √(u2 - 1)) + C | |u| ≥ 1 |
- Select the inverse trigonometric function you want to integrate.
- Enter multiplier k and linear values a and b.
- Provide the plotting range using start and end x values.
- Enter one sample x value for direct antiderivative evaluation.
- Click Calculate Integral to view the formula, values, and graph.
- Use the CSV and PDF buttons to export the displayed result block.
- Check the domain rule before trusting any sample or range value.
1) What does this calculator compute?
It computes antiderivatives for inverse trigonometric expressions in the form k·f-1(ax+b). It also evaluates the antiderivative at chosen points, estimates change across a range, and plots the resulting curve.
2) Which inverse trigonometric functions are included?
The tool supports arcsin, arccos, arctan, arccot, arcsec, and arccsc. Each option uses a matching closed-form antiderivative and applies domain checks before showing numeric results.
3) Why must the coefficient a not equal zero?
The method substitutes u = ax+b, which requires division by a. If a equals zero, the chosen general formula breaks because the substitution step and scale factor k/a become undefined.
4) Why do some inputs show outside domain?
Inverse trigonometric functions are not defined for every real input. For example, arcsin and arccos need values from -1 to 1, while arcsec and arccsc need absolute values at least 1.
5) Does the answer include the constant of integration?
Yes. The symbolic antiderivative always ends with +C. Numeric evaluations compare antiderivative values at specific x points, so the constant cancels when you inspect the change across a range.
6) What does F(x end) - F(x start) represent?
It represents the net change in the antiderivative across the selected interval. When both endpoints are valid, it matches the definite integral value from the Fundamental Theorem of Calculus.
7) Are the graph values exact or approximate?
The graph is numerical and uses many sampled points across your chosen range. The symbolic formula is exact, while displayed decimal values are rounded for readability and export.
8) Can I export the result after calculation?
Yes. The page includes CSV and PDF export buttons inside the result section. They capture the calculated summary so you can keep a simple record for study, checking, or sharing.