Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx

검증된 단계별 안내

Recognize that the integral is \( \int \frac{\tan^{-1}(x)}{x^2} \, dx \), where \( \tan^{-1}(x) \) is the inverse tangent function, also written as \( \arctan(x) \).

Consider using integration by parts, since the integrand is a product of functions: one involving \( \arctan(x) \) and the other involving \( \frac{1}{x^2} \). Set \( u = \arctan(x) \) and \( dv = \frac{1}{x^2} dx \).

Compute \( du \) and \( v \): - \( du = \frac{1}{1+x^2} dx \) because the derivative of \( \arctan(x) \) is \( \frac{1}{1+x^2} \). - \( v = \int x^{-2} dx = -x^{-1} = -\frac{1}{x} \).

Apply the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] Substitute the expressions for \( u, v, du \) to rewrite the integral.

Simplify the resulting integral and evaluate it using the table of integrals if necessary, focusing on the integral \( \int \frac{1}{x(1+x^2)} dx \) that appears after substitution.

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주요 개념

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Inverse Trigonometric Functions

Inverse trigonometric functions, like arctan(x), are the inverses of the standard trig functions and return angles given a ratio. Understanding their properties and derivatives is essential for integrating expressions involving these functions.

Integration Techniques

Integration techniques such as integration by parts or substitution are often required to solve integrals involving products or compositions of functions, especially when standard formulas are not directly applicable.

Use of Integral Tables

Integral tables provide formulas for common integrals, including those involving inverse trig functions. Knowing how to locate and apply these formulas can simplify the evaluation of complex integrals.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ tan^(-1)(x) / x² dx