Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / √(1 - x²)

검증된 단계별 안내

Recognize that the integral \( \int \frac{dx}{\sqrt{1 - x^2}} \) resembles the derivative of an inverse trigonometric function, specifically \( \arcsin x \) or \( \arccos x \).

Recall the derivative formula: \( \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \). This suggests that the integral is related to \( \arcsin x \).

Set up the integral using the substitution method if needed, but in this case, direct recognition is sufficient since the integrand matches the derivative of \( \arcsin x \).

Write the integral as \( \int \frac{dx}{\sqrt{1 - x^2}} = \arcsin x + C \), where \( C \) is the constant of integration.

Verify the result by differentiating \( \arcsin x + C \) to confirm it returns the original integrand \( \frac{1}{\sqrt{1 - x^2}} \).

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주요 개념

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

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots of expressions like 1 - x² by substituting x with a trigonometric function such as sin(θ). This transforms the integral into a trigonometric integral that is easier to evaluate.

Inverse Trigonometric Functions

Integrals involving expressions like 1/√(1 - x²) often result in inverse trigonometric functions, specifically arcsin(x). Recognizing this form helps directly identify the antiderivative without complex manipulation.

Basic Integration Techniques

Understanding fundamental integration rules and recognizing standard integral forms allows for efficient evaluation. For example, knowing that ∫ dx/√(1 - x²) = arcsin(x) + C avoids unnecessary steps.

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ dx / √(1 - x²)