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Trigonometric Substitution Calculator

Choose the radical form and enter a to get the recommended trig substitution.

How it is calculated

You pick one of three radical forms and enter a (a positive number). The tool does not integrate anything; it returns the standard substitution for that form. √(a² − x²): x = a·sin θ, radical a·cos θ, dx = a·cos θ dθ. √(a² + x²): x = a·tan θ, radical a·sec θ, dx = a·sec² θ dθ. √(x² − a²): x = a·sec θ, radical a·tan θ, dx = a·sec θ·tan θ dθ. The value a is the square root of the constant under the radical, so 25 − x² means a = 5. Zero or negative a gives no result. The number a is written in the formulas with only the decimals it needs and no thousands separator, so 1000 appears as 1000. The radical is shown without absolute values or a restriction on the range of θ.

Worked example

Take the integrand with √(2.5² + x²), so a = 2.5 and the form is a² + x². Because the sign between the terms is plus, the tangent case applies. Substitution: x = 2.5·tan θ. Then 2.5² + x² = 2.5²(1 + tan² θ) = 2.5² sec² θ, so the radical simplifies to 2.5·sec θ. Differentiating x gives dx = 2.5·sec² θ dθ. You can now rewrite the whole integral in θ.

Try this example →

Common mistakes

  • Entering the number under the root instead of its square root. For √(9 − x²) you must enter a = 3, not 9.
  • Choosing the form by the look of the sign. x² − a² (secant) and a² − x² (sine) are different cases, and the order of the terms matters.
  • Forgetting that the tool gives only the substitution, the radical and dx. You still have to rewrite the rest of the integrand, integrate in θ and convert back to x with a right triangle.
  • Using the tool for a coefficient such as √(4x² + 9). Rewrite it first (for example with u = 2x) so that it matches one of the three forms.
  • Ignoring the range of θ. The simplification of the radical assumes cos θ, sec θ or tan θ is positive on the chosen interval.

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.

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Comments (1)

  • Editorial team

    This tool only names the substitution; it does not integrate. For the form with x squared minus a squared and a of 3, it returns x = 3·sec θ, radical 3·tan θ and dx = 3·sec θ·tan θ dθ. Once you integrate, remember to return to x using a reference triangle. Which integral are you stuck on, and which of the three forms does it contain?

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