Integral of Trig Functions Calculator

Calculator Form

Enter values in radians. Use bounds only for definite integration.

Function type sin(ax+b) cos(ax+b) tan(ax+b) cot(ax+b) sec(ax+b) csc(ax+b) sec(ax+b)tan(ax+b) csc(ax+b)cot(ax+b) sec²(ax+b) csc²(ax+b) sin(ax+b)cos(ax+b) sin²(ax+b) cos²(ax+b)

Integration mode Indefinite Definite

Multiplier k

Inner coefficient a

Inner shift b

Chosen constant C

Evaluation point x

Lower bound

Upper bound

Calculate Integral

Plotly Graph

The graph shows the integrand and one antiderivative branch with C = 0.

Example Data Table

Formula Used

This calculator applies substitution to expressions built from ax+b. When the inside term is linear, the antiderivative keeps the standard trig pattern and adds a scaling factor of 1/a.

• ∫sin(ax+b)dx = -cos(ax+b)/a + C
• ∫cos(ax+b)dx = sin(ax+b)/a + C
• ∫tan(ax+b)dx = -ln|cos(ax+b)|/a + C
• ∫cot(ax+b)dx = ln|sin(ax+b)|/a + C
• ∫sec(ax+b)dx = ln|sec(ax+b)+tan(ax+b)|/a + C
• ∫csc(ax+b)dx = -ln|csc(ax+b)+cot(ax+b)|/a + C
• ∫sec²(ax+b)dx = tan(ax+b)/a + C
• ∫csc²(ax+b)dx = -cot(ax+b)/a + C
• ∫sec(ax+b)tan(ax+b)dx = sec(ax+b)/a + C
• ∫csc(ax+b)cot(ax+b)dx = -csc(ax+b)/a + C

For products and powers, the calculator uses common identities such as sin(u)cos(u), sin²(u), and cos²(u). Those forms are reduced before integration so the output stays practical and readable.

How to Use This Calculator

1. Choose the trigonometric form that matches your integrand.
2. Enter the outside multiplier k.
3. Enter the inner values a and b for ax+b.
4. Set a chosen constant C if you want a numeric evaluation of the antiderivative.
5. Enter an x value to evaluate the antiderivative numerically.
6. Select definite mode and enter bounds when you need an area value.
7. Press Calculate Integral to show the result above the form.
8. Use the export buttons to save the current result as CSV or PDF.

About This Integral of Trig Functions Calculator

Why this tool is useful

Integrating trigonometric expressions becomes easier when the structure is recognized early. This calculator is designed for students who want a faster way to test patterns, confirm substitution steps, and review final antiderivatives without switching between many examples. It handles common single trig forms, derivative-pair patterns, and selected product or power identities in one place.

What it calculates

The tool works with expressions built around a linear inner angle, usually written as ax+b. That structure appears in many algebra, calculus, and exam problems. Once the inside expression is linear, the chain rule tells us that the final integral should include a scaling factor related to a. The calculator applies that factor automatically, then returns a symbolic antiderivative, a numeric function value at a chosen point, and a definite integral when bounds are supplied.

How the graph helps

The included graph gives a visual connection between the original integrand and one antiderivative branch. This is helpful for checking whether the result behaves as expected. For example, when the original expression stays positive on an interval, the antiderivative should increase there. When the trig expression changes sign, the antiderivative slope changes too. That visual relationship reinforces the meaning of integration instead of showing only a formula line.

Study and verification value

This page also helps with classroom practice, worksheets, and revision sessions. You can compare different coefficient choices, inspect the formula used, and export a clean record of the calculation. The example table supports quick reference, while the step list highlights the substitution idea behind the answer. Used regularly, the calculator can strengthen recognition of standard trig integrals and reduce avoidable algebra mistakes during manual work.

FAQs

1. What does this calculator solve?

It finds antiderivatives for common trigonometric forms, including sin, cos, tan, cot, sec, csc, selected products, and selected squared expressions built from ax+b.

2. Are the angle inputs in degrees or radians?

This page uses radians. Enter all shifts, bounds, and evaluation points in radians so the formulas and graph stay mathematically consistent.

3. Why does the answer include a factor of 1/a?

That factor comes from substitution. When u=ax+b, then du=a dx, so the integral must be scaled by 1/a after rewriting dx.

4. Can I compute a definite integral too?

Yes. Choose definite mode, then enter both bounds. The calculator evaluates the antiderivative at each bound and subtracts the results.

5. Why do some inputs give an undefined result?

Functions such as tan, sec, cot, and csc are undefined at certain angles. If your evaluation point or bounds hit those locations, the result cannot be returned there.

6. What does the chosen constant C do?

The symbolic result always keeps +C. Your entered value of C is used only for the numeric antiderivative value shown at the selected x point.

7. What does the graph represent?

The graph plots the original integrand and one antiderivative branch with C set to zero. It helps you compare slope behavior and overall shape.

8. Can I export my result?

Yes. After calculating, you can download the current result as CSV or PDF for saving, revision, or sharing with classmates.