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Trigonometric Substitution Reference cheat sheet - grade 11-12

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Calculus Grade 11-12

Trigonometric Substitution Reference Cheat Sheet

A printable reference covering trigonometric substitutions, radical forms, triangle back-substitution, identities, and integral setup for grades 11-12.

TopicsIntegralsTrigonometryAlgebraCalculusStudy Guide
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Trigonometric substitution is a calculus technique used to simplify integrals containing radicals such as a2−x2\sqrt{a^2 - x^2}, a2+x2\sqrt{a^2 + x^2}, and x2−a2\sqrt{x^2 - a^2}. This cheat sheet helps students choose the correct substitution, rewrite the differential, and convert the integral into a trigonometric form. It is especially useful when completing problems involving areas, arc length, and inverse trigonometric results.

The key idea is to match the radical expression to a Pythagorean identity. For a2−x2\sqrt{a^2 - x^2} use x=asin⁡θx = a\sin \theta, for a2+x2\sqrt{a^2 + x^2} use x=atan⁡θx = a\tan \theta, and for x2−a2\sqrt{x^2 - a^2} use x=asec⁡θx = a\sec \theta. After integrating in terms of θ\theta, use a right triangle or inverse trig relation to convert the answer back to xx.

Key Facts

  • For radicals of the form a2−x2\sqrt{a^2 - x^2}, use x=asin⁡θx = a\sin \theta and dx=acos⁡θ dθdx = a\cos \theta\,d\theta.
  • For radicals of the form a2+x2\sqrt{a^2 + x^2}, use x=atan⁡θx = a\tan \theta and dx=asec⁡2θ dθdx = a\sec^2 \theta\,d\theta.
  • For radicals of the form x2−a2\sqrt{x^2 - a^2}, use x=asec⁡θx = a\sec \theta and dx=asec⁡θtan⁡θ dθdx = a\sec \theta\tan \theta\,d\theta.
  • The identity 1−sin⁡2θ=cos⁡2θ1 - \sin^2 \theta = \cos^2 \theta simplifies a2−x2\sqrt{a^2 - x^2} after the substitution x=asin⁡θx = a\sin \theta.
  • The identity 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta simplifies a2+x2\sqrt{a^2 + x^2} after the substitution x=atan⁡θx = a\tan \theta.
  • The identity sec⁡2θ−1=tan⁡2θ\sec^2 \theta - 1 = \tan^2 \theta simplifies x2−a2\sqrt{x^2 - a^2} after the substitution x=asec⁡θx = a\sec \theta.
  • Back-substitution uses a right triangle built from the equation such as sin⁡θ=xa\sin \theta = \frac{x}{a}, tan⁡θ=xa\tan \theta = \frac{x}{a}, or sec⁡θ=xa\sec \theta = \frac{x}{a}.
  • Always convert the final antiderivative back to xx unless the problem specifically asks for an answer in terms of θ\theta.

Vocabulary

Trigonometric substitution
A method for rewriting an integral using a trigonometric expression for xx so that a radical becomes easier to simplify.
Radical form
The expression under a square root, such as a2−x2a^2 - x^2, a2+x2a^2 + x^2, or x2−a2x^2 - a^2, that determines which substitution to use.
Differential
The rewritten form of dxdx, such as dx=acos⁡θ dθdx = a\cos \theta\,d\theta, needed after substituting for xx.
Pythagorean identity
A trigonometric identity such as 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta that comes from the Pythagorean theorem.
Back-substitution
The process of converting an answer from θ\theta back to xx using a triangle or inverse trigonometric relationship.
Reference triangle
A right triangle built from the substitution equation to find trig functions in terms of xx.

Common Mistakes to Avoid

  • Using x=atan⁡θx = a\tan \theta for a2−x2\sqrt{a^2 - x^2} is wrong because that radical matches the identity 1−sin⁡2θ=cos⁡2θ1 - \sin^2 \theta = \cos^2 \theta, so x=asin⁡θx = a\sin \theta is the standard choice.
  • Forgetting to replace dxdx is wrong because the integral must be completely rewritten in terms of θ\theta, including the differential such as dx=asec⁡2θ dθdx = a\sec^2 \theta\,d\theta.
  • Dropping the square root simplification too early is wrong because a2cos⁡2θ\sqrt{a^2\cos^2 \theta} simplifies to a∣cos⁡θ∣a|\cos \theta|, and the chosen interval usually justifies writing acos⁡θa\cos \theta.
  • Leaving the final answer in terms of θ\theta is incomplete when the original integral uses xx, so use a reference triangle or inverse trig relation to back-substitute.
  • Mixing identities such as 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta and 1−sin⁡2θ=cos⁡2θ1 - \sin^2 \theta = \cos^2 \theta leads to incorrect simplification, so match the identity to the radical form first.

Practice Questions

  1. 1 Choose the correct trigonometric substitution and rewrite dxdx for ∫116−x2 dx\int \frac{1}{\sqrt{16 - x^2}}\,dx.
  2. 2 Use trigonometric substitution to set up the integral ∫x2x2+9 dx\int \frac{x^2}{\sqrt{x^2 + 9}}\,dx in terms of θ\theta.
  3. 3 Evaluate ∫1x2x2−4 dx\int \frac{1}{x^2\sqrt{x^2 - 4}}\,dx using an appropriate trigonometric substitution.
  4. 4 Explain why x=asec⁡θx = a\sec \theta is better than x=asin⁡θx = a\sin \theta for simplifying x2−a2\sqrt{x^2 - a^2}.