Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / √(1 - 4x²) from 0 to 1/(2√2)

검증된 단계별 안내

Recognize that the integral has the form \( \int \frac{2}{\sqrt{1 - 4x^2}} \, dx \), which resembles the standard integral \( \int \frac{dx}{\sqrt{1 - u^2}} = \arcsin(u) + C \).

Make a substitution to simplify the integral. Let \( u = 2x \), so that \( du = 2 \, dx \) or equivalently \( dx = \frac{du}{2} \).

Rewrite the integral in terms of \( u \): \( \int \frac{2}{\sqrt{1 - 4x^2}} \, dx = \int \frac{2}{\sqrt{1 - u^2}} \cdot \frac{du}{2} = \int \frac{1}{\sqrt{1 - u^2}} \, du \).

Evaluate the integral \( \int \frac{1}{\sqrt{1 - u^2}} \, du = \arcsin(u) + C \).

Substitute back \( u = 2x \) and apply the definite integral limits: from \( x = 0 \) to \( x = \frac{1}{2\sqrt{2}} \), which correspond to \( u = 0 \) to \( u = \frac{1}{\sqrt{2}} \). Then compute \( \arcsin(u) \) at these limits and find the difference.

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주요 개념

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

Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the antiderivative at the upper and lower bounds and subtracting these values to find the exact accumulated quantity.

Integration of Functions Involving Square Roots

Integrals containing expressions like √(1 - a²x²) often relate to inverse trigonometric functions. Recognizing these forms allows the use of substitution or standard integral formulas involving arcsin or arccos to simplify the integration process.

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a standard form. For example, setting u = 2x can help rewrite the integral into a form involving √(1 - u²), making it easier to integrate using known formulas.

Evaluate the integrals in Exercises 1–14.∫ (2 dx) / √(1 - 4x²) from 0 to 1/(2√2)