7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
51. ∫ x²/√(4 + x²) dx

검증된 단계별 안내

Step 1: Recognize that the integral involves a square root of the form √(a² + x²), which suggests using the trigonometric substitution x = 2tan(θ). This substitution simplifies the square root expression. Let a = 2.

Step 2: Substitute x = 2tan(θ) into the integral. Compute dx = 2sec²(θ)dθ and replace x² with (2tan(θ))² = 4tan²(θ). The square root √(4 + x²) becomes √(4 + 4tan²(θ)) = √(4sec²(θ)) = 2sec(θ).

Step 3: Rewrite the integral in terms of θ using the substitution. The integral becomes ∫ (4tan²(θ) / (2sec(θ))) * 2sec²(θ)dθ. Simplify the expression by canceling terms where possible.

Step 4: Simplify the integral further to ∫ 4tan²(θ)sec(θ)dθ. Use trigonometric identities, such as tan²(θ) = sec²(θ) - 1, to simplify the integrand.

Step 5: Solve the integral in terms of θ. After integrating, convert back to the original variable x using the substitution x = 2tan(θ) and the relationship tan(θ) = x/2.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

13m

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 x = a tan(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² + x²), allowing for easier integration.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is crucial when using trigonometric substitution, as it allows us to express one trigonometric function in terms of another. For example, if we substitute x = 2 tan(θ), we can use this identity to simplify the resulting expressions involving √(4 + x²) during integration.

Integration Techniques

Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and trigonometric substitution. Understanding these techniques is essential for solving complex integrals, as they provide strategies to transform and simplify the integrand. Mastery of these methods enables students to tackle a wide range of problems in calculus effectively.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.51. ∫ x²/√(4 + x²) dx