6.13 Integrating Using Trigonometric Substitution

• Steps to use trigonometric substitution in integration

  1️⃣ Identify when to use trigonometric substitution

  2️⃣ Choose the appropriate trigonometric substitution

  3️⃣ Evaluate the integral using the substitution

  4️⃣ Rewrite the result in terms of the original variable
• Trigonometric substitution is used when an integral contains expressions of the form \sqrt{a^{2} - x^{2}}
• Match the expression with its trigonometric substitution

  $\sqrt{a^{2} - x^{2}}$ ↔️ $x =$$a\sin(\theta)$

  \sqrt{a^{2} + x^{2}} ↔️ $x =$$a\tan(\theta)$

  $\sqrt{x^{2} - a^{2}}$ ↔️ $x =$$a\sec(\theta)$
• Trigonometric substitutions simplify integrals by eliminating square roots using trigonometric identities.
• To choose the correct trigonometric substitution, you must consider the form of the expression under the square root.
• Match the substitution with its corresponding expression

  $x =$$a\sin(\theta)$ ↔️ $\sqrt{a^{2} - x^{2}}$

  $x =$$a\tan(\theta)$ ↔️ \sqrt{a^{2} + x^{2}}

  $x =$$a\sec(\theta)$ ↔️ $\sqrt{x^{2} - a^{2}}$
• Steps to evaluate an integral using trigonometric substitution

  1️⃣ Identify the expression under the square root

  2️⃣ Choose the appropriate substitution

  3️⃣ Rewrite the integral in terms of $\theta$

  4️⃣ Evaluate the simplified integral

  5️⃣ Rewrite the result in terms of the original variable
• Trigonometric substitution is only applicable to integrals with square roots.
• Match the expression with its corresponding substitution

  $\sqrt{a^{2} - x^{2}}$ ↔️ $x =$$a\sin(\theta)$

  \sqrt{a^{2} + x^{2}} ↔️ $x =$$a\tan(\theta)$

  $\sqrt{x^{2} - a^{2}}$ ↔️ $x =$$a\sec(\theta)$
• What trigonometric substitution should be used for \sqrt{a^{2} + x^{2}}?


  $x =$$a\tan(\theta)$
• Trigonometric substitutions simplify integrals by eliminating square roots through trigonometric identities.
• Match the expression with its appropriate substitution and trigonometric identity:

  $\sqrt{a^{2} - x^{2}}$ ↔️ $x =$$a\sin(\theta)$, $1 - \sin^{2}(\theta) =$$\cos^{2}(\theta)$

  \sqrt{a^{2} + x^{2}} ↔️ $x =$$a\tan(\theta)$, $1 +$$\tan^{2}(\theta) =$$\sec^{2}(\theta)$

  $\sqrt{x^{2} - a^{2}}$ ↔️ $x =$$a\sec(\theta)$, $\sec^{2}(\theta) - 1 =$$\tan^{2}(\theta)$
• The expression $\sqrt{a^{2} - x^{2}}$ requires the substitution x = a\sin(\theta)</latex> and uses the identity $\cos^{2}(\theta) =$$1 - \sin^{2}(\theta)$ to eliminate the square
• The trigonometric substitution $x =$$a\sec(\theta)$ is used to simplify expressions of the form $\sqrt{x^{2} - a^{2}}$.
• When choosing a trigonometric substitution, the most important factor is the form of the expression under the square root.
• What trigonometric identity is used when simplifying $\sqrt{a^{2} - x^{2}}$ with $x =$$a\sin(\theta)$?


  $\cos^{2}(\theta) =$$1 - \sin^{2}(\theta)$
• What trigonometric identity is used when simplifying \sqrt{a^{2} + x^{2}} with $x =$$a\tan(\theta)$?


  $\sec^{2}(\theta) =$$1 +$$\tan^{2}(\theta)$
• The substitution $x =$$a\sec(\theta)$ simplifies $\sqrt{x^{2} - a^{2}}$ using the identity $\tan^{2}(\theta) =$$\sec^{2}(\theta) - 1$, which eliminates the square root.
• Trigonometric substitutions simplify integrals by applying trigonometric identities to eliminate square roots.
• Steps to evaluate the integral $\int \sqrt{4 - x^{2}} \, dx$ using trigonometric substitution:


  1️⃣ Identify the form $\sqrt{a^{2} - x^{2}}$ with $a =$$2$ and choose $x =$$2\sin(\theta)$

  2️⃣ Find $dx =$$2\cos(\theta) \, d\theta$

  3️⃣ Substitute into the integral and simplify using $\cos^{2}(\theta) =$$1 - \sin^{2}(\theta)$

  4️⃣ Use the identity $\cos^{2}(\theta) =$$\frac{1}{2}(1 + \cos(2\theta))$

  5️⃣ Express $\theta$ and $\sin(2\theta)$ in terms of $x$

  6️⃣ Write the final answer: $2\arcsin\left(\frac{x}{2}\right) +$$\frac{x}{2}\sqrt{4 - x^{2}} +$$C$
• What is $\sin(2\theta)$ in terms of $x$ when $x =$$2\sin(\theta)$?


  $\frac{x}{2}\sqrt{4 - x^{2}}$
• When rewriting the result of a trigonometric substitution, you must express $\theta$ in terms of $x$.