The Trigonometric Substitution Calculator helps evaluate integrals containing square-root expressions that match common quadratic patterns. Trigonometric identities can turn the radical into a simpler expression that is easier to integrate.
When Trigonometric Substitution Helps
Typical patterns include √(a²−x²), √(a²+x²), and √(x²−a²). Common substitutions are x=a sinθ for the first pattern, x=a tanθ for the second, and x=a secθ for the third. The appropriate choice depends on the expression and domain.
How the Method Works
The substitution changes both x and dx. A trigonometric identity then simplifies the square root. After integration, the result may need to be converted back from θ to the original variable.
Example
For an integral containing √(a²−x²), let x=a sinθ. Then dx=a cosθ dθ and √(a²−x²)=a cosθ over a suitable interval. The radical is replaced by a trigonometric expression, often making the integral much easier to handle.
Using the Calculator
Enter the expression exactly as requested and verify that the selected substitution matches its structure. If the tool returns an inverse trigonometric term, check the domain before simplifying the final expression.
Important Check
Trigonometric substitution is a technique, not a universal requirement. Simpler algebraic substitution or another integration method may be better for some expressions. The final result should be differentiated when possible to verify the antiderivative.