Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
46. ∫ 1/√(1 - 2x²) dx
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
4:57 minutesProblem 8.4.44
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
44. ∫ 1/√(16 + 4x²) dx
Verified step by step guidance1
Step 1: Recognize that the integral involves a square root of the form √(a² + u²), which suggests using the trigonometric substitution u = a * tan(θ). Here, identify a = 4 and u = 2x, so let 2x = 4 * tan(θ).
Step 2: Substitute u = 2x = 4 * tan(θ). Compute dx by differentiating: dx = d(2x) = 4 * sec²(θ) dθ.
Step 3: Replace √(16 + 4x²) using the trigonometric identity 1 + tan²(θ) = sec²(θ). Substituting, √(16 + 4x²) becomes √(16 * sec²(θ)) = 4 * sec(θ).
Step 4: Rewrite the integral in terms of θ using the substitutions: ∫ 1/√(16 + 4x²) dx = ∫ (1 / (4 * sec(θ))) * (4 * sec²(θ) dθ). Simplify the expression to ∫ sec(θ) dθ.
Step 5: Integrate ∫ sec(θ) dθ using the standard formula for the integral of sec(θ), which is ln|sec(θ) + tan(θ)| + C. Finally, back-substitute θ using the original substitution 2x = 4 * tan(θ) to express the result in terms of x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as sine or tangent, the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² + x²), a² - x², or x² - a².
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Identifying the Right Triangle
When using trigonometric substitution, it is essential to visualize the relationship between the variable and the trigonometric function through a right triangle. For example, in the integral ∫ 1/√(16 + 4x²) dx, we can set x = 2tan(θ), which leads to a right triangle where the opposite side is 2x and the adjacent side is 4. This helps in determining the appropriate trigonometric identities to apply.
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Integration Techniques
After performing the substitution, the integral often requires the application of various integration techniques, such as basic integration rules or further substitutions. In the case of the integral ∫ 1/√(16 + 4x²) dx, after substituting and simplifying, one may need to integrate a trigonometric function, which can involve recognizing standard integral forms or using integration by parts if necessary.
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