Calculus I · Unit 3A · lesson

Trigonometric Substitution

Learn Trigonometric Substitution through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.

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Section overview

Computing integrals

What this section is building

Learn Trigonometric Substitution through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.

Notice

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

Decide

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

Avoid

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

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

\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 $1-\sin^2\theta=\cos^2\theta$ , $1+\tan^2\theta=\sec^2\theta$ , and $\sec^2\theta-1=\tan^2\theta$ .

Worked example

A circular radical

Evaluate

\int\frac{dx}{\sqrt{9-x^2}}.

Let $x=3\sin\theta$ , so $dx=3\cos\theta\,d\theta$ and $\sqrt{9-x^2}=3\cos\theta$ on the chosen interval. Then

\int d\theta=\theta+C=\arcsin(x/3)+C.

Interactive checku3a-trig-sub-01

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

Your work stays on this device. No account or AI grader is used.

Show hint

Match $x^2+a^2$ to $1+\tan^2\theta$ .

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

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

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Trigonometric Substitution

Explanation

Use a trigonometric identity to simplify a radical

Standard patterns are:

\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^2\theta=\cos^2\theta

1+\tan^2\theta=\sec^2\theta

, and

\sec^2\theta-1=\tan^2\theta

Worked example

A circular radical

Evaluate

\int\frac{dx}{\sqrt{9-x^2}}.

Let $x=3\sin\theta$ , so $dx=3\cos\theta\,d\theta$ and $\sqrt{9-x^2}=3\cos\theta$ on the chosen interval. Then

\int d\theta=\theta+C=\arcsin(x/3)+C.

Interactive checku3a-trig-sub-01

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

Your work stays on this device. No account or AI grader is used.

Show hint

Match $x^2+a^2$ to $1+\tan^2\theta$ .

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

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.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Trigonometric Substitution

Related resources

Original instruction with traceable references.

Trigonometric Substitution

Related resources

Original instruction with traceable references.