Calculator Inputs
The page stays in a single vertical flow, while the calculator fields respond as 3 columns on large screens, 2 on medium, and 1 on mobile.
Formula Used
These standard identities and antiderivative rules power the calculator.
- ∫ sin(u) dx = −cos(u)/u′ + C for a linear inner angle.
- ∫ cos(u) dx = sin(u)/u′ + C for a linear inner angle.
- ∫ tan(u) dx = −ln|cos(u)|/u′ + C.
- ∫ cot(u) dx = ln|sin(u)|/u′ + C.
- ∫ sec²(u) dx = tan(u)/u′ + C.
- ∫ csc²(u) dx = −cot(u)/u′ + C.
- ∫ sec(u)tan(u) dx = sec(u)/u′ + C.
- ∫ csc(u)cot(u) dx = −csc(u)/u′ + C.
- sin²(u) = (1 − cos(2u))/2.
- cos²(u) = (1 + cos(2u))/2.
- ∫ sin(u)cos(u) dx = sin²(u)/(2u′) + C for linear u.
How to Use This Calculator
- Select the trigonometric integral pattern that matches your expression.
- Enter the coefficient a, inner coefficient b, and phase shift c.
- Choose Indefinite for an antiderivative or Definite for a bounded evaluation.
- When using definite mode, enter lower and upper bounds in radians.
- Set the plot range and sample count for the graph.
- Press Submit and Solve to show the result above the form.
- Use the export buttons to download the current result as CSV or PDF.
Example Data Table
| Example | Input Pattern | Method | Antiderivative Output |
|---|---|---|---|
| 1 | 2·sin(3x) | u-substitution | −(2/3)cos(3x) + C |
| 2 | 4·cos(2x + 1) | u-substitution | 2·sin(2x + 1) + C |
| 3 | 5·sec²(x) | standard derivative pair | 5·tan(x) + C |
| 4 | 3·sin²(2x) | power-reduction identity | (3/2)x − (3/8)sin(4x) + C |
| 5 | 6·sin(x)cos(x) | pattern recognition | 3·sin²(x) + C |
FAQs
1. What integrals can this tool solve?
It handles linear-angle sine, cosine, tangent, cotangent, secant-squared, cosecant-squared, secant-tangent, cosecant-cotangent, sine-squared, cosine-squared, and sine-cosine product forms. It returns indefinite antiderivatives and definite values when the interval stays on one continuous branch.
2. Are the angles entered in degrees?
No. The calculator uses radians. Standard derivative and antiderivative rules are naturally written in radians, so entering degree values directly will produce different results unless you convert them first.
3. Why is my definite integral not computed?
Some trig functions have vertical asymptotes. If your interval crosses a point where sine or cosine becomes zero in the denominator, the ordinary definite integral is not safe to evaluate, so the calculator stops and shows a warning.
4. Does the tool return exact algebra or only decimals?
It returns an exact symbolic antiderivative string for the chosen pattern and also provides a decimal definite result when that bounded evaluation is valid.
5. What happens when b equals zero?
The inner angle becomes constant. In that case, the trig expression turns into a constant value, and the calculator automatically reduces the integral to the simpler form kx + C.
6. What does the graph show?
The Plotly chart displays both the original integrand and one antiderivative branch across your chosen x-range. This helps you compare slope behavior and see where discontinuities affect the expression.
7. Can I export the result?
Yes. The result panel includes CSV and PDF export buttons. The exported file contains the selected pattern, numeric inputs, computed antiderivative, method summary, formula note, and definite value status.
8. Is this a general symbolic integration engine?
No. It is a focused advanced calculator for common trigonometric integral families with linear inner angles. That design keeps the output clear, fast, and reliable for teaching, checking, and practice.