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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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