Evaluate the integrals in Exercises 41–60.
57. ∫(from 1 to 2)cosh(ln t)/t dt

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Recognize the integral to evaluate: \(\int_{1}^{2} \frac{\cosh(\ln t)}{t} \, dt\).

Recall the definition of hyperbolic cosine: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\). Substitute \(x = \ln t\) to rewrite the integrand.

Rewrite the integrand as \(\frac{1}{t} \cdot \frac{e^{\ln t} + e^{-\ln t}}{2} = \frac{1}{2t} (t + \frac{1}{t})\) because \(e^{\ln t} = t\) and \(e^{-\ln t} = \frac{1}{t}\).

Simplify the expression inside the integral to \(\frac{1}{2t} (t + \frac{1}{t}) = \frac{1}{2} \left(1 + \frac{1}{t^{2}}\right)\).

Split the integral into two simpler integrals: \(\frac{1}{2} \int_{1}^{2} 1 \, dt + \frac{1}{2} \int_{1}^{2} \frac{1}{t^{2}} \, dt\), then proceed to integrate each term separately.

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주요 개념

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

Hyperbolic Cosine Function (cosh)

The hyperbolic cosine function, cosh(x), is defined as (e^x + e^(-x))/2. It is similar to the cosine function but based on exponential functions. Understanding its properties helps simplify expressions involving cosh, especially when combined with logarithmic arguments.

Logarithmic Functions and Their Properties

The natural logarithm function, ln(t), is the inverse of the exponential function e^x. Key properties, such as ln(ab) = ln(a) + ln(b) and ln(t^n) = n ln(t), allow simplification of integrands involving ln(t). Recognizing how ln(t) interacts with other functions is crucial for integration.

Integration Techniques and Substitution

Integration by substitution is a method to simplify integrals by changing variables. Here, substituting u = ln(t) transforms the integral into a more manageable form. Mastery of substitution and recognizing when to apply it is essential for evaluating integrals involving composite functions.

Evaluate the integrals in Exercises 41–60.57. ∫(from 1 to 2)cosh(ln t)/t dt