Trigonometric Substitution

Trigonometric substitutions are a specific type of u-substitution and rely heavily on techniques developed for u-substitutions. Trigonometric substitution uses Pythagorean identities to turn a difficult algebraic square root expression involving sums or differences of squares into a single, clean trigonometric term. This converts a hard calculus problem into a simpler trigonometric integral that can be solved and then reversed using a right triangle.

Contents

Standard Types

The entire trigonometric substitution technique comes down to one idea:

Calculus Fact

Match the radical to the correct trigonometric identity.

When you see a radical involving x² and a², there are only three standard types:

$\sqrt{x^2-a^2}\,, \sqrt{x^2+a^2}\,, \sqrt{x^2-a^2}$

Each Type Has Its Own Substitution

Type I

$\sqrt{a^2-x^2}\,\,\Rightarrow x\,\,=\,\,a\sin \left( \theta \right), dx\,\,=\,\,a\cos \left( \theta \right) d\theta$

Type II

$\sqrt{x^2+a^2}\,\,\Rightarrow x\,\,=\,\,a\tan \left( \theta \right) , dx\,\,=\,\,a\sec \left( \theta \right) d\theta$

Type III

$\sqrt{x^2-a^2}\,\,\Rightarrow x\,\,=\,\,a\sec \left( \theta \right) , dx\,\,=\,\,a\sec \left( \theta \right) \tan \left( \theta \right) d\theta$

where

x = The original variable of integration.
a = A positive constant appearing in the expression under the radical.
θ = The new variable introduced through the trigonometric substitution.

Trigonometric Substitution

Trig substitution is used when an integral contains a radical involving x² and a².
Determine if the Type I, II or III based on the radicand and then turn the radical into its corresponding trigonometric identity.
After substituting, integrate with respect to θ.
Never leave the final answer in θ. Use a reference triangle to convert back to x.

Trig Substitution Key Facts

The expression under the square root tells you which substitution to use.
Type I matches sin²(θ) + cos²(θ) = 1
Type II matches 1 + tan²(θ) = sec²(θ)
Type III matches sec²(θ) – 1 = tan²(θ)
Each substitution creates a triangle that converts the final answer back to x.

Key Limitations and Constraints

Trigonometric substitution is limited by the need to restrict variable domains (𝜃) to ensure inverse functions are defined and single-valued. It requires specific algebraic forms (like $the square root of a squared minus x squared end-root$ ), can be less efficient than simpler methods like 𝑢-substitution, and requires careful handling when converting limits of integration. To make trigonometric functions one-to-one so they have inverses, the substitution variable (𝜃) must be restricted.

Domain Restrictions (Restrictions of Inverse Functions):

For $x equals a sine theta$ , 𝜃 is restricted to $open bracket negative pi / 2 comma pi / 2 close bracket$ .
For $x equals a tangent theta$ , 𝜃 is restricted to $open paren negative pi / 2 comma pi / 2 close paren$ .
For $x equals a secant theta$ , 𝜃 is restricted to $open bracket 0 comma pi / 2 close paren union open paren pi / 2 comma pi close bracket$ .

Definite Integral Limits

When calculating definite integrals, you must convert the 𝑥-limits to 𝜃-limits, which involves solving for 𝜃 using inverse functions (e.g., $theta equals arc sine open paren x / a close paren$ ), rather than simply substituting 𝑥 into a $u open paren x close paren$ formula.

Alternative Techniques

Many integrals solvable by trig substitution can be solved more quickly with simpler substitution or other techniques, such as partial fractions.

Domain of Integration

The substitution must be valid over the entire interval of integration. For example, if the substitution results in a division by zero or an imaginary number (like $the square root of negative 1 end-root$ ), the substitution is invalid for that interval.

Reference Triangles

For an explanation of how each of the standard types relates to a triangle, see 3.3 Trigonometric Substitution – Calculus Volume 2.

**Figure 3.4** A reference triangle can help express the trigonometric functions evaluated at 𝜃 in terms of 𝑥
.

**Figure 3.7** A reference triangle can be constructed to express the trigonometric functions evaluated at 𝜃 in terms of 𝑥
.

**Figure 3.9** Use the appropriate reference triangle to express the trigonometric functions evaluated at 𝜃 in terms of 𝑥
.

Practice Problems

Use the following websites to practice trigonometric substitution.

“7.3E: Exercises for Trigonometric Substitution.” LibreTexts, March 12, 2019. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_211_Calculus_II/Chapter_7%3A_Techniques_of_Integration/7.3%3A_Trigonometric_Substitution/7.3E%3A_Exercises_for_Trigonometric_Substitution.

Dawkins, Paul. “Calculus II.” Trig Substitutions (Practice Problems), Paul’s Online Notes, November 11, 2022. https://tutorial.math.lamar.edu/problems/calcii/trigsubstitutions.aspx.

Hidegkuti, Marta. “Trigonometric Substitutions.” Lecture Notes. teaching.martahidegkuti.com. https://teaching.martahidegkuti.com/shared/lnotes/6_calculus/integral/trigsubstitutions/trigsubstitutions.pdf.

References

“Integration with Trigonometric Substitution.” StudyPug, Accessed July 15, 2026. https://www.studypug.com/calculus-help/trigonometric-substitution/?view=read.

[ ℰ ] Strang, Gilbert, Edwin “Jed” Herman, Gilbert Strang, and Edwin “Jed” Herman. “3.3 Trigonometric Substitution – Calculus Volume 2.” OpenStax, March 30, 2016. https://openstax.org/books/calculus-volume-2/pages/3-3-trigonometric-substitution.

Strang, Gilbert, and Edwin “Jed” Herman. “7.3: Trigonometric Substitution.” LibreTexts, July 11, 2016. https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/07%3A_Techniques_of_Integration/7.03%3A_Trigonometric_Substitution.

“Trig Substitution Table: √(a²−x²), √(a²+x²), √(x²−a²).” Mathwords, Accessed July 15, 2026. https://www.mathwords.com/t/trig_substitution.htm.

[ ℰ ] Woody, Brian M. “Trig Substitution Explained.” Woody Calculus, July 7, 2026. https://www.brianwoody.com/trig-substitution-calculus-2-three-type-system/.

Additional Reading

Dawkins, Paul. “Calculus II.” Trig Substitutions. Paul’s Online Notes, October 16, 2023. https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx.

[ ℰ ] “Trigonograph.” Mathematical Mysteries, January 1, 2024. https://mathematicalmysteries.org/trigonograph/.

“Trigonograph” is this author’s name for a geometric diagram that illustrates a trigonometric relation or concept, ideally in a way that makes things “obvious”. The term is a back-construction of the word-playful trigonography, which could be taken to mean “the art of trigonometric visualization”.

“Trigonometric Substitutions.” SFU. Accessed July 15, 2026. https://www.sfu.ca/math-coursenotes/Math%20158%20Course%20Notes/sec_Trig_Sub.html.

Videos

Calculus 2: Trigonometric Substitution (Video #3)

Examples applying trigonometric substitution in order to evaluate indefinite and definite integrals. Three cases explained with multiple examples; uses of the Pythagorean identities, additional integration techniques, and nuanced mathematical skills.

Evaluating Integrals Using Trigonometric Substitution (Trig Sub) | Math with Professor V

Integration Using Trigonometric Substitution (Trig Sub) | Calculus 2 | Math with Professor V

Four examples demonstrating evaluating integrals using trigonometric substitution or “trig sub”. Review of integration techniques, pythagorean identities, and how to simplify and clean up your final answer.

Evaluating Trigonometric Integrals | Powers of Sine, Cosine, Tangent & More! | Math with Professor V

Updated video lecture on how to evaluate trigonometric integrals. Clear outlining of the various cases, how to use trigonometric identities and u-substitution. Powers of sine, cosine, tangent, and secant explained.

Mastering TRIG SUB for Integration! Step by Step Explanation | Math with Professor V

Evaluating integrals using TRIG SUB–as essential skill that all Calculus students need to master! In this video, I go over several examples demonstrating the three different cases and types of trig subs that we use to evaluate integrals and solve them step by step. These examples vary in difficulty, with the final problem being quite challenging. If you can solve all the exercises from this video again after on your own, then you’re well on your way to being a trig sub pro! So enjoy!

Changing Limits of Integration for U-Sub vs. Trig Sub: Easy Explanation! Math with Professor V

Having difficulty switching your limits of integration for definite integrals when doing u-sub and trig sub? This is a common issue for many students, and often there is a lot of confusion since there are differences in the process depending on the integration technique. In this video I’ll work through several examples to help clarify HOW to switch your limits of integration for u-sub vs. trig sub, and help you become a PRO at this before your next exam! Enjoy!

[ ℰ ] This exceptional reference is highly recommended for your consideration.

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