Trigonometric Substitution Calculator

About this calculator

Trigonometric substitution is an integration technique for expressions containing a²−x², a²+x², or x²−a², replacing x with a trig function of a new variable θ so the expression under the radical collapses into a single squared trig term via a Pythagorean identity — a²−x² becomes a·cosθ with x = a·sinθ, a²+x² becomes a·secθ with x = a·tanθ, and x²−a² becomes a·tanθ with x = a·secθ. This calculator identifies the matching case from your chosen radical form and value of a, returning the correct substitution, the simplified radical, and dx.

The technique is a core topic in integral calculus courses, typically taught after u-substitution and integration by parts, and is essential for evaluating integrals that appear in physics and engineering problems involving circles, ellipses, and hyperbolic geometry — students commonly use a tool like this to check which of the three substitution cases applies before working through the algebra by hand.