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Calculus I · Unit 3A · lesson

Trigonometric Substitution

Concept

Learning objectives

Match radical patterns to trigonometric identities and return to the original variable using a reference triangle.

Trigonometric Substitution

Explanation

Use a trigonometric identity to simplify a radical

Expressions such as a2−x2\sqrt{a^2-x^2}, a2+x2\sqrt{a^2+x^2}, and x2−a2\sqrt{x^2-a^2} resemble the Pythagorean identities for sine, tangent, and secant. Trigonometric substitution chooses a new variable so that the radical collapses to a simpler trigonometric expression. The method is less about trigonometry itself than about replacing a difficult algebraic geometry with a familiar right-triangle identity.

The substitution must respect the domain and the sign of the square root. Drawing a reference triangle makes the return to xx concrete and prevents inverse-trigonometric confusion. Because this method can create substantial algebra, it should not be the first reflex for every radical. Simplify first and check whether an ordinary substitution is available before deploying the full trigonometric apparatus.

Standard patterns are:

a2−x2: x=asin⁡θ,a2+x2: x=atan⁡θ,x2−a2: x=asec⁡θ.\sqrt{a^2-x^2}:\ x=a\sin\theta, \qquad \sqrt{a^2+x^2}:\ x=a\tan\theta, \qquad \sqrt{x^2-a^2}:\ x=a\sec\theta.

The substitutions are chosen because 1−sin⁡2θ=cos⁡2θ1-\sin^2\theta=\cos^2\theta, 1+tan⁡2θ=sec⁡2θ1+\tan^2\theta=\sec^2\theta, and sec⁡2θ−1=tan⁡2θ\sec^2\theta-1=\tan^2\theta.

Worked example

A circular radical

Evaluate

∫dx9−x2.\int\frac{dx}{\sqrt{9-x^2}}.

Let x=3sin⁡θx=3\sin\theta, so dx=3cos⁡θ dθdx=3\cos\theta\,d\theta and 9−x2=3cos⁡θ\sqrt{9-x^2}=3\cos\theta on the chosen interval. Then

∫dθ=θ+C=arcsin⁡(x/3)+C.\int d\theta=\theta+C=\arcsin(x/3)+C.
Interactive checku3a-trig-sub-01

Which substitution is natural for x2+25\sqrt{x^2+25}?

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Show hint

Match x2+a2x^2+a^2 to 1+tan⁡2θ1+\tan^2\theta.

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Reference triangles for trig substitution. Connect radical forms to right triangles and substitution choices.
Read this graph as text

Reference triangles for trig substitution. Three right triangles encode x=a sin theta, x=a tan theta, and x=a sec theta. Connect radical forms to right triangles and substitution choices. State domain restrictions and the role of absolute values.

Every relationship in reference triangles for trig substitution uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.

Why it matters: Connect radical forms to right triangles and substitution choices.

Visual study

Reference triangles for trig substitution. Connect radical forms to right triangles and substitution choices.

After the explanation

Use the section idea

Reading lens

Choose an integration method from the integrand's structure, then verify the result by differentiation.

Mental model

Substitution reverses a chain rule, parts reverses a product rule, and algebraic or trigonometric rewrites expose a recognizable derivative pattern.

Decision

Simplify first; look for an inner derivative; then consider parts, identities, trigonometric substitution, or partial fractions in a deliberate order.

Common trap

Choosing a method by surface appearance, transforming only part of the differential, or accepting a more complicated integral than the one you started with.

Check yourself

Can you name the derivative rule being reversed and differentiate the final answer back to the integrand?

Source & rights

Original instruction with traceable references.

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