7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
46. ∫ 1/√(1 - 2x²) dx

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Step 1: Recognize the integral involves a square root of the form √(1 - 2x²). This suggests using a trigonometric substitution to simplify the expression. Specifically, use the substitution x = (1/√2)sin(θ), which ensures that 1 - 2x² becomes 1 - sin²(θ).

Step 2: Compute dx in terms of θ. Differentiating x = (1/√2)sin(θ) gives dx = (1/√2)cos(θ)dθ.

Step 3: Substitute x = (1/√2)sin(θ) and dx = (1/√2)cos(θ)dθ into the integral. The square root √(1 - 2x²) becomes √(1 - sin²(θ)), which simplifies to cos(θ). The integral now becomes ∫(1/√2)cos(θ)/(cos(θ))dθ.

Step 4: Simplify the integral. The cos(θ) terms cancel out, leaving ∫(1/√2)dθ. This is a straightforward integral of a constant.

Step 5: Integrate the constant and back-substitute θ in terms of x using the original substitution x = (1/√2)sin(θ). Use the relationship sin(θ) = √2x to express the final 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 cosine, the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(1 - x²) or √(x² - a²).

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is fundamental in trigonometric substitution as it allows us to express one trigonometric function in terms of another, facilitating the simplification of integrals. For example, if we let x = sin(θ), then √(1 - x²) becomes cos(θ), which can simplify the integral significantly.

Integral Evaluation

Integral evaluation is the process of finding the antiderivative of a function, which can often be achieved through various techniques, including substitution, integration by parts, or trigonometric identities. In the context of trigonometric substitution, once the integral is transformed into a trigonometric form, it can be evaluated using standard integral formulas. After finding the antiderivative, it is essential to revert back to the original variable to express the final answer.

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