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Questions and Answers
What is the main purpose of using trigonometric substitution in integration?
What is the main purpose of using trigonometric substitution in integration?
To remove radicals from the integrand.
For an integrand containing $\sqrt{a^2 - x^2}$, what is the appropriate trigonometric substitution for x?
For an integrand containing $\sqrt{a^2 - x^2}$, what is the appropriate trigonometric substitution for x?
$x = a \sin(\theta)$
If you encounter $\sqrt{a^2 + x^2}$ in an integral, what trigonometric substitution should you use for x?
If you encounter $\sqrt{a^2 + x^2}$ in an integral, what trigonometric substitution should you use for x?
$x = a \tan(\theta)$
What trigonometric substitution is suitable when the integrand includes $\sqrt{x^2 - a^2}$?
What trigonometric substitution is suitable when the integrand includes $\sqrt{x^2 - a^2}$?
When using the substitution $x = a \sin(\theta)$, what does $\sqrt{a^2 - x^2}$ become?
When using the substitution $x = a \sin(\theta)$, what does $\sqrt{a^2 - x^2}$ become?
After substituting $x = a \tan(\theta)$, to simplify an integral, what is $\sqrt{a^2 + x^2}$ equal to?
After substituting $x = a \tan(\theta)$, to simplify an integral, what is $\sqrt{a^2 + x^2}$ equal to?
If you use the substitution $x = a \sec(\theta)$, what expression does $\sqrt{x^2 - a^2}$ simplify to?
If you use the substitution $x = a \sec(\theta)$, what expression does $\sqrt{x^2 - a^2}$ simplify to?
What should you remember to include after evaluating an indefinite integral?
What should you remember to include after evaluating an indefinite integral?
What is the derivative dx when you use the substitution $x = a \sin(\theta)$?
What is the derivative dx when you use the substitution $x = a \sin(\theta)$?
Flashcards
Trigonometric Substitution
Trigonometric Substitution
A technique for evaluating integrals by replacing the original variable with a trigonometric function to remove radicals from the integrand.
Case 1: √(a² - x²)
Case 1: √(a² - x²)
Use when the integrand contains √(a² - x²). Substitute x = asin(θ), then dx = acos(θ) dθ.
Case 2: √(a² + x²)
Case 2: √(a² + x²)
Use when the integrand contains √(a² + x²). Substitute x = atan(θ), then dx = asec²(θ) dθ.
Case 3: √(x² - a²)
Case 3: √(x² - a²)
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Steps for Trig Substitution
Steps for Trig Substitution
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Integrals with √(a² - x²)
Integrals with √(a² - x²)
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Integrals with √(a² + x²)
Integrals with √(a² + x²)
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Integrals with √(x² - a²)
Integrals with √(x² - a²)
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Study Notes
- Trigonometric substitution is an integration technique to eliminate radicals from the integrand
- It involves replacing the original variable with a trigonometric function
- This simplifies the integral and allows it to be solved with trig identities
- It is useful when the integrand contains √(a² - x²), √(a² + x²), or √(x² - a²)
When to Use Trigonometric Substitution
- Useful when the integrand contains square roots: √(a² - x²), √(a² + x²), or √(x² - a²)
- 'a' represents a constant term.
- 'x' represents the variable of integration.
Three Main Cases for Trigonometric Substitution
Case 1: √(a² - x²)
- For integrands containing √(a² - x²), substitute x = a * sin(θ)
- Consequently, dx = a * cos(θ) dθ
- √(a² - x²) simplifies to a * cos(θ)
- The identity 1 - sin²(θ) = cos²(θ) is useful
Case 2: √(a² + x²)
- For integrands containing √(a² + x²), substitute x = a * tan(θ)
- Therefore, dx = a * sec²(θ) dθ
- √(a² + x²) converts to a * sec(θ)
- The identity 1 + tan²(θ) = sec²(θ) is useful
Case 3: √(x² - a²)
- For integrands containing √(x² - a²), substitute x = a * sec(θ)
- Then, dx = a * sec(θ) * tan(θ) dθ
- √(x² - a²) simplifies to a * tan(θ)
- The identity sec²(θ) - 1 = tan²(θ) is useful
Procedure for Trigonometric Substitution
- Select the appropriate substitution based on the square root's form
- Substitute for x and dx in the original integral.
- Simplify the integral using trigonometric identities.
- Evaluate the resulting trigonometric integral.
- Convert the result back to the original variable x.
Example: Evaluating an Integral Using Trigonometric Substitution
- Evaluation of ∫√(4 - x²) dx
- Integrand contains √(4 - x²), matching √(a² - x²) with a = 2
- Substitute x = 2 * sin(θ), dx = 2 * cos(θ) dθ
- Substitute into the integral: ∫√(4 - (2sinθ)²) * (2cosθ) dθ = ∫√(4 - 4sin²θ) * (2cosθ) dθ
- Simplify: ∫2cosθ * 2cosθ dθ = 4∫cos²θ dθ
- Apply the identity cos²(θ) = (1 + cos(2θ))/2: 4∫(1 + cos(2θ))/2 dθ = 2∫(1 + cos(2θ)) dθ
- Integrate: 2[θ + (1/2)sin(2θ)] + C = 2θ + sin(2θ) + C
- Convert back to x: x = 2sin(θ), resulting in sin(θ) = x/2, θ = arcsin(x/2)
- Further convert sin(2θ) = 2sin(θ)cos(θ) = 2(x/2)√(1 - (x/2)²) = (x/2)√(4 - x²)
- Final answer: 2arcsin(x/2) + (x/2)√(4 - x²) + C
Tips and Common Mistakes
- Choose the correct trigonometric substitution based on the form of the square root
- Simplify the integral carefully using trigonometric identities
- Use the initial substitution to express θ in terms of x when converting back to the original variable
- Include the constant of integration, C, after evaluating the indefinite integral
- Take care with algebraic manipulations and trigonometric identities to avoid errors
Integrals Involving √(a² - x²)
- Substitute x = a sin θ
- dx = a cos θ dθ
- √(a² - x²) = a cos θ
- The identity 1 - sin²θ = cos²θ is useful
Integrals Involving √(a² + x²)
- Substitute x = a tan θ
- dx = a sec² θ dθ
- √(a² + x²) = a sec θ
- The identity 1 + tan²θ = sec²θ is useful
Integrals Involving √(x² - a²)
- Substitute x = a sec θ
- dx = a sec θ tan θ dθ
- √(x² - a²) = a tan θ
- The identity sec²θ - 1 = tan²θ is useful
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