Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to 3 of (1 / √(9 − x²)) dx

검증된 단계별 안내

Recognize that the integral \( \int_0^3 \frac{1}{\sqrt{9 - x^2}} \, dx \) is an improper integral because the integrand involves a square root in the denominator that becomes zero at the upper limit \( x = 3 \). This means the function approaches infinity there, so we need to treat the integral as a limit.

Rewrite the integral as a limit: \( \lim_{t \to 3^-} \int_0^t \frac{1}{\sqrt{9 - x^2}} \, dx \). This allows us to evaluate the integral over \( [0, t] \) where \( t < 3 \) and then take the limit as \( t \) approaches 3 from the left.

Recall the antiderivative formula for \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left( \frac{x}{a} \right) + C \). In this problem, \( a = 3 \), so the antiderivative is \( \arcsin\left( \frac{x}{3} \right) + C \).

Evaluate the definite integral from 0 to \( t \) using the antiderivative: \( \int_0^t \frac{1}{\sqrt{9 - x^2}} \, dx = \arcsin\left( \frac{t}{3} \right) - \arcsin(0) \). Since \( \arcsin(0) = 0 \), this simplifies to \( \arcsin\left( \frac{t}{3} \right) \).

Finally, take the limit as \( t \to 3^- \): \( \lim_{t \to 3^-} \arcsin\left( \frac{t}{3} \right) \). This will give the value of the improper integral.

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주요 개념

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

Improper Integrals

Improper integrals involve integrals with infinite limits or integrands that approach infinity within the interval. To evaluate them, limits are used to handle points where the function is undefined or unbounded, ensuring the integral converges to a finite value.

Integration of Functions Involving Square Roots

Integrals containing expressions like √(a² - x²) often relate to inverse trigonometric functions. Recognizing these forms allows the use of substitution or standard integral formulas, such as the arcsine function, to simplify and evaluate the integral.

Trigonometric Substitution

Trigonometric substitution replaces variables with trigonometric functions to simplify integrals involving radicals like √(a² - x²). For example, substituting x = a sin θ transforms the integral into a trigonometric form that is easier to integrate.

Evaluate the improper integrals in Exercises 53–62.∫ from 0 to 3 of (1 / √(9 − x²)) dx