Trigonometric Substitution Calculator

The Trigonometric Substitution Calculator helps evaluate integrals containing expressions such as √(a² − x²), √(a² + x²), or √(x² − a²) by replacing x with a suitable trigonometric expression. Use the tool to work through the calculation and review the result.

Definite
Indefinite

with respect to

Upper value

Lower value

Common substitutions

For √(a² − x²), use x = a sin θ.
For √(a² + x²), use x = a tan θ.
For √(x² − a²), use x = a sec θ.

Worked example

For ∫√(9 − x²) dx, let x = 3 sin θ. Then dx = 3 cos θ dθ and √(9 − x²) becomes 3 cos θ. The radical is removed, leaving a trigonometric integral that can be evaluated and then converted back to x.

Practical tip

Choose the substitution that matches the expression under the square root, and keep track of the corresponding differential and triangle relationships when returning to the original variable.

Frequently Asked Questions FAQ

Which substitution fits √(a²−x²)?

A common choice is x=a sin θ, which changes the radical into a trigonometric expression involving cos θ.

What is an example of trigonometric substitution?

For ∫√(9−x²)dx, use x=3 sin θ. Then dx=3 cos θ dθ and the radical becomes 3 cos θ.

Why is the substitution reversed at the end?

After integrating in θ, the result must be expressed in the original variable x unless the problem is intentionally left in terms of θ.

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