Evaluate the integrals in Exercises 53–76.
55. ∫dx/(17+x²)

검증된 단계별 안내

Recognize that the integral \( \int \frac{dx}{17 + x^2} \) is of the form \( \int \frac{dx}{a^2 + x^2} \), which is a standard integral that results in an arctangent function.

Identify \( a^2 = 17 \), so \( a = \sqrt{17} \). This allows us to rewrite the integral as \( \int \frac{dx}{(\sqrt{17})^2 + x^2} \).

Recall the formula for the integral: \( \int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan\left( \frac{x}{a} \right) + C \), where \( C \) is the constant of integration.

Apply the formula by substituting \( a = \sqrt{17} \) into the expression, giving \( \frac{1}{\sqrt{17}} \arctan\left( \frac{x}{\sqrt{17}} \right) + C \).

Write the final answer as the integral evaluated, including the constant of integration \( C \).

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주요 개념

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

Integration of Rational Functions

This involves integrating functions expressed as ratios of polynomials. Recognizing the form of the integrand helps in choosing the appropriate method, such as substitution or partial fractions, to simplify and evaluate the integral.

Integral of the Form ∫dx/(a² + x²)

Integrals of the form ∫dx/(a² + x²) have a standard solution: (1/a) arctangent(x/a) + C. This formula is essential for evaluating integrals where the denominator is a sum of a constant squared and the variable squared.

Arctangent Function and Its Properties

The arctangent function, denoted arctan(x), is the inverse of the tangent function. It arises naturally in integrals involving 1/(a² + x²), and understanding its derivative and behavior is key to correctly applying integration formulas.

Evaluate the integrals in Exercises 53–76.55. ∫dx/(17+x²)