Decision guideCalculus IIAdvanced12 min read

Choosing the right trigonometric substitution

Match a quadratic radical to a Pythagorean identity so the square root simplifies instead of becoming worse.

a^2-x^2,\quad a^2+x^2,\quad x^2-a^2

Decision guide

A practical comparison that turns a vague choice into a repeatable test.

Reviewed July 11, 2026

Pattern mapStart here

\sqrt{a^2-x^2}:x=a\sin\theta\quad \sqrt{a^2+x^2}:x=a\tan\theta\quad \sqrt{x^2-a^2}:x=a\sec\theta

Choose the substitution whose Pythagorean identity collapses the radical to one trig function.

The three structural patterns

For a² − x², sine creates 1 − sin²θ = cos²θ. For a² + x², tangent creates 1 + tan²θ = sec²θ. For x² − a², secant creates sec²θ − 1 = tan²θ.

Complete the square first when the quadratic is shifted or scaled.

Track the differential and domain

Every substitution changes dx as well as the radical. Choose a θ-interval that makes the square-root simplification consistent with a nonnegative radical.

A reference triangle gives a reliable back-substitution without memorizing inverse identities.

Know when a simpler substitution exists

A radical alone does not force trig substitution. If the derivative of the expression under the root is already present, ordinary u-substitution is shorter and clearer.

Worked exampleUse the a² − x² pattern

Evaluate ∫ dx/√(9 − x²).

x=3\sin\theta,\qquad dx=3\cos\theta\,d\theta

Match a = 3.

\sqrt{9-9\sin^2\theta}=3\cos\theta

Use the Pythagorean identity on a suitable interval.

\int d\theta=\theta+C

The factors cancel completely.

Result

\boxed{\arcsin(x/3)+C}

Watch for

Common mistakes

Choosing tangent for the a² − x² pattern.
Forgetting to transform dx.
Back-substituting without a triangle or a valid inverse relation.

Keep

Three takeaways

Match the sign pattern to a Pythagorean identity.
Complete the square before classifying.
Check whether u-substitution is simpler first.