Trigonometric Substitution

7 Techniques of Integration

7.3 Trigonometric Substitution

Overview of Trigonometric Substitution

• Integral Forms: Trigonometric substitution is invoked particularly when evaluating integrals that involve square roots of forms such as $ext{a}^2 - x^2$, $ext{a}^2 + x^2$, and $x^2 - ext{a}^2$.

• Purpose: The substitution allows simplification of integrals to a form that can be easily integrated using trigonometric identities.

Trigonometric Substitution Technique

Substitution Selection

• Choosing a Substitution:

  • For expressions like $ext{a}^2 - x^2$, use $x = ext{a} ext{sin} heta$.

  • For expressions like $ext{a}^2 + x^2$, use $x = ext{a} ext{tan} heta$.

  • For expressions like $x^2 - ext{a}^2$, use $x = ext{a} ext{sec} heta$.

Formulating the Substitution

• Identity Utilization:

  • The identity $ext{sin}^2 heta + ext{cos}^2 heta = 1$ allows transformation of the radical expressions effectively:

  • If $x = ext{a} ext{sin} heta$, then $ext{sqrt}( ext{a}^2 - x^2) <br>ightarrow ext{sqrt}( ext{a}^2(1 - ext{sin}^2 heta)) = ext{a} ext{cos} heta$.

• Differential Change: When substituting, compute dy:

  • E.g., $dx = ext{a} ext{cos} heta d heta$ for appropriate substitutions.

Substitution Variants

Inverse Substitution

• Definition: Inverse substitution involves replacing variables where the new variable is a function of the old variable, e.g., using $x = ext{a} ext{sin} heta$.

• General Form: If replacing in the form $x = g(t)$, we must assume $g$ is one-to-one to facilitate the substitution process.

Trigonometric Substitutions Table

• 
  

• Common Substitutions:
  

  Radical Expression	Substitute
  $ext{sqrt}(a^2 - x^2)$	$x = a ext{sin} θ$
  $ext{sqrt}(a^2 + x^2)$	$x = a ext{tan} θ$
  $ext{sqrt}(x^2 - a^2)$	$x = a ext{sec} θ$
  Worked Examples

  Example 1: Evaluating an Integral

  • Integral Setup: Evaluate $rac{9 - x^2}{2} dx$ by substituting $x = 3 ext{sin} θ$.

  • Differential Calculation: Compute $dx = 3 ext{cos} θ dθ$.

  • Integral Transformation:

    • The integral becomes:
      $rac{1}{2} imes ext{Integral} ext{of} ext{another expression determined by} ext{trigonometric identities}$.

  Example 2: Area of an Ellipse

  • Area Setup: To find the area enclosed by an ellipse, solve for y in terms of x, then observe the symmetry:

    • Total Area: $A = 4$ times the area in the first quadrant.

  • Integral Calculation: Use substitution $x = a ext{sin} θ$ leading to efficient evaluation of the area.

  • Final Area Equation: The area with semi-minor axis $b$ and semi-major axis $a$ derives as:
    $A = ext{πab}$.

  Example 3: Integral of a Composite Function

  • Integral Components: Using $x = 2 ext{tan} θ$ leads to simplification:

  • Expression Conversion: Transform expression in terms of sin and cos for easy integration.

  Note on Hyperbolic Substitution

  • Sometimes hyperbolic forms can be used instead:

    • E.g., for expressions like $x^2 - a^2$ can substitute $x = a ext{cosh} t$.

  • Summary of Advantages: Trigonometric identities are often more familiar than hyperbolic, making trigonometric substitutions typically more straightforward.

  Conclusion

  • Trigonometric substitutions provide effective means to evaluate complex integrals involving radicals by transforming variables and utilizing identities strategically to simplify calculations. The understanding of identities and the setup of substitutions is crucial in leading to accurate results in integrals involving circular or elliptical forms.