Evaluate the integrals in Exercises 1–6.
∫ (arcsin x)² dx

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Recognize that the integral involves the square of the inverse sine function, \((\arcsin x)^2\), which suggests using integration by parts to simplify the expression.

Set up integration by parts by choosing \(u = (\arcsin x)^2\) and \(dv = dx\). Then compute \(du\) and \(v\): - Differentiate \(u\) using the chain rule: \(du = 2 \arcsin x \cdot \frac{1}{\sqrt{1 - x^2}} \, dx\). - Integrate \(dv\) to get \(v = x\).

Apply the integration by parts formula: \[\int u \, dv = uv - \int v \, du,\] which becomes \[\int (\arcsin x)^2 \, dx = x (\arcsin x)^2 - 2 \int \frac{x \arcsin x}{\sqrt{1 - x^2}} \, dx.\]

Focus on the remaining integral \(\int \frac{x \arcsin x}{\sqrt{1 - x^2}} \, dx\). Use substitution to simplify it: let \(t = \arcsin x\), so that \(x = \sin t\) and \(dx = \cos t \, dt\). Rewrite the integral in terms of \(t\).

After substitution, simplify the integral and solve it using standard integration techniques (such as integration by parts again or recognizing standard integrals). Finally, substitute back \(t = \arcsin x\) to express the answer in terms of \(x\).

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주요 개념

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

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv is crucial, especially when integrating functions like (arcsin x)².

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin x, are the inverses of the standard trigonometric functions. Understanding their derivatives and properties is essential for integration, as these functions often appear in integrals and require special techniques.

Reduction of Powers in Integration

When integrating powers of functions, it is often helpful to use reduction formulas or rewrite the integrand to simplify the integral. For (arcsin x)², expressing the square in a form suitable for integration by parts or substitution aids in solving the integral efficiently.

Evaluate the integrals in Exercises 1–6.∫ (arcsin x)² dx