Verify the integration formulas in Exercises 37–40.
39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C

검증된 단계별 안내

Identify the integral to verify: \(\int x \coth^{-1}(x) \, dx\) and the proposed formula: \(\frac{(x^2 - 1)}{2} \coth^{-1}(x) + \frac{x}{2} + C\).

Use integration by parts, where you let \(u = \coth^{-1}(x)\) and \(dv = x \, dx\). Then compute \(du\) and \(v\):

\[ u = \coth^{-1}(x) \implies du = \frac{-1}{1 - x^2} \, dx, \quad dv = x \, dx \implies v = \frac{x^2}{2} \]

Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so write:

\[ \int x \coth^{-1}(x) \, dx = \frac{x^2}{2} \coth^{-1}(x) - \int \frac{x^2}{2} \cdot \left( \frac{-1}{1 - x^2} \right) dx \]

Simplify the integral inside and solve it step-by-step, then combine all terms to verify that the result matches the given formula.

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주요 개념

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

Inverse Hyperbolic Cotangent Function (coth⁻¹(x))

The inverse hyperbolic cotangent function, coth⁻¹(x), is the inverse of the hyperbolic cotangent. It is defined for |x| > 1 and can be expressed using logarithms. Understanding its domain and properties is essential for integrating expressions involving coth⁻¹(x).

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals. This method is often used when integrating products like x * coth⁻¹(x), where one function is easily differentiable and the other integrable.

Verification of Integration Formulas

Verifying an integration formula involves differentiating the proposed antiderivative to check if it yields the original integrand. This process confirms the correctness of the integral and constant of integration, ensuring the formula is valid for the given function.

Verify the integration formulas in Exercises 37–40.39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C