Calculator Input
This tool returns an antiderivative formula, endpoint values, a definite integral result, and a graph of both the integrand and antiderivative.
Example Data Table
| Example | Integral Form | Antiderivative | Example Definite Value |
|---|---|---|---|
| 1 | ∫ sinh(2x) dx | cosh(2x)/2 + C | From 0 to 1: 1.381098 |
| 2 | ∫ tanh(3x+1) dx | ln(cosh(3x+1))/3 + C | From 0 to 2: 1.670864 |
| 3 | ∫ sech²(4x) dx | tanh(4x)/4 + C | From -1 to 1: 0.499665 |
| 4 | ∫ sinh(x)cosh(x) dx | sinh²(x)/2 + C | From 0 to 1: 0.690549 |
Formula Used
The calculator uses substitution with u = ax + b. That converts each integral into a standard hyperbolic form.
- ∫ sinh(u) du = cosh(u) + C
- ∫ cosh(u) du = sinh(u) + C
- ∫ tanh(u) du = ln(cosh(u)) + C
- ∫ coth(u) du = ln|sinh(u)| + C
- ∫ sech(u) du = 2 arctan(tanh(u/2)) + C
- ∫ csch(u) du = ln|tanh(u/2)| + C
- ∫ sech²(u) du = tanh(u) + C
- ∫ csch²(u) du = -coth(u) + C
- ∫ sinh²(u) du = sinh(2u)/4 - u/2 + C
- ∫ cosh²(u) du = sinh(2u)/4 + u/2 + C
How to Use This Calculator
- Select the hyperbolic integral type from the dropdown.
- Enter the scale factor k, coefficient a, and shift b.
- Enter lower and upper limits for the definite integral.
- Set the graph interval to inspect function behavior.
- Click Calculate Integral to view formulas, values, steps, and the graph.
- Use the export buttons to save your calculation summary.
FAQs
1. What does this calculator solve?
It solves common hyperbolic integrals, including sinh, cosh, tanh, sech, csch, squared forms, and selected product forms with linear inner expressions.
2. Does it show indefinite and definite results?
Yes. It returns an antiderivative formula and also evaluates the definite integral using the chosen lower and upper limits.
3. Why must a not equal zero?
The substitution method divides by the inner derivative. If a equals zero, the transformation no longer matches the standard formulas used here.
4. Why can some values become undefined?
Functions like coth and csch are undefined where sinh(ax+b) equals zero. The calculator marks those points as undefined in values or graphs.
5. What does the graph display?
The graph shows both the original integrand and one antiderivative. That helps compare growth, symmetry, turning behavior, and interval changes.
6. Can I export the result?
Yes. The calculator includes CSV and PDF export buttons for the current result summary, including settings and computed values.
7. Are the formulas exact?
The symbolic antiderivative follows exact standard identities. Numeric endpoint values and graph coordinates are displayed in rounded decimal form.
8. Who can use this tool best?
It is useful for students, tutors, exam practice, and quick verification when working with substitution and hyperbolic integral identities.