Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x arctan(x) dx

검증된 단계별 안내

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

Recognize that this integral is a product of two functions: \(x\) and \(\arctan(x)\), which suggests using integration by parts.

Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Choose \(u = \arctan(x)\) (since its derivative is simpler) and \(dv = x \, dx\).

Compute \(du\) and \(v\): differentiate \(u\) to get \(du = \frac{1}{1 + x^2} \, dx\), and integrate \(dv\) to get \(v = \frac{x^2}{2}\).

Apply the integration by parts formula: write the integral as \(\frac{x^2}{2} \arctan(x) - \int \frac{x^2}{2} \cdot \frac{1}{1 + x^2} \, dx\), then simplify the integrand and prepare to evaluate the remaining integral.

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주요 개념

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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 appropriately simplifies the integral, especially when one function becomes simpler upon differentiation.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arctan(x), are the inverses of trigonometric functions and have specific derivatives and integrals. Understanding their properties and derivatives, such as d/dx[arctan(x)] = 1/(1+x²), is essential for integrating expressions involving these functions.

Using a Table of Integrals

A table of integrals provides formulas for common integrals, saving time and effort. It is useful for recognizing standard forms and applying known results directly, especially when integrals involve special functions or combinations that are difficult to integrate from first principles.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ x arctan(x) dx