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

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Identify the integral to solve: \(\int x \arcsin x \, dx\).

Use integration by parts, where you let \(u = \arcsin x\) and \(dv = x \, dx\).

Compute \(du\) by differentiating \(u\): \(du = \frac{1}{\sqrt{1 - x^2}} \, dx\), and compute \(v\) by integrating \(dv\): \(v = \frac{x^2}{2}\).

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

Simplify the remaining integral \(\int \frac{x^2}{2 \sqrt{1 - x^2}} \, dx\) and consider substitution methods to evaluate it.

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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 u and dv wisely simplifies the integral, especially when one function becomes simpler upon differentiation.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arcsin(x), are the inverses of the standard trigonometric functions. Understanding their derivatives and integrals is essential, as arcsin(x) has a derivative of 1/√(1 - x²), which often appears in integration problems involving these functions.

Algebraic Manipulation in Integration

Algebraic manipulation involves rewriting integrals into simpler or more recognizable forms. This can include substitution, simplifying expressions, or breaking integrals into parts, which helps in applying integration techniques effectively and solving complex integrals step-by-step.

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