Trigonometric Substitution

The central idea

Trigonometric substitution uses identities to simplify quadratic radicals: for $\sqrt{a^2-x^2}$ set $x=a\sin\theta$; for $\sqrt{a^2+x^2}$ set $x=a\tan\theta$; for $\sqrt{x^2-a^2}$ set $x=a\sec\theta$. Choose an angle interval so signs and inverse substitutions are controlled.

Definitions, hypotheses, and notation

Hyperbolic substitutions can sometimes avoid absolute-value complications, but the trig forms align with standard introductory identities. Completing the square first extends the method to quadratics such as $x^2+4x+13=(x+2)^2+9$. The shifted variable, not the original $x$, then matches the substitution.

For definite integrals, transform bounds to angles and remain in $\theta$, or back-substitute before applying $x$-bounds. As in ordinary substitution, mixing coordinate systems in the bounds is invalid.

Conceptual meaning

Each substitution converts a sum or difference of squares into a perfect trig square using $1-\sin^2=\cos^2$, $1+\tan^2=\sec^2$, or $\sec^2-1=\tan^2$. A reference triangle translates the final trig expression back to $x$.

A dependable method and decision rule

1. Complete the square or factor constants to match a standard radical.
2. Choose the corresponding trig substitution and compute $dx$.
3. Simplify the radical with an identity, respecting absolute values.
4. Integrate in $\theta$.
5. Back-substitute using an inverse trig function or labelled triangle, then check the original domain.

Fully worked example

Graphical or geometric meaning

For $x=3\sin\theta$, a right triangle has hypotenuse $3$, opposite side $x$, and adjacent side $\sqrt{9-x^2}$. The triangle converts any remaining sine, cosine, or tangent back into algebraic expressions.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate the final expression.
• Verify the chosen angle interval makes radical simplification valid.
• Compare against a known inverse-trig derivative when applicable.

The radical determines the triangle

Use $x=a\sin\theta$ for $\sqrt{a^2-x^2}$, $x=a\tan\theta$ for $\sqrt{a^2+x^2}$, and $x=a\sec\theta$ for $\sqrt{x^2-a^2}$. Choose a range for $\theta$ that makes the square root and inverse relation unambiguous; otherwise an absolute value may be lost. Simplify with the corresponding Pythagorean identity, integrate in $\theta$, and return to $x$ using a labeled triangle or algebraic relation. For a definite integral, transformed bounds can eliminate back-substitution. Differentiate the final expression on the stated domain. The substitution is a domain-aware change of variable, not simply a visual replacement of symbols.

Practice

1. Choose a substitution for $\sqrt{16-x^2}$.
2. Choose one for $\sqrt{x^2+25}$.
3. Choose one for $\sqrt{x^2-9}$.

Connections and next steps

• Trigonometric Integrals
• Substitution Rule
• Partial Fractions