Evaluate the integrals in Exercises 111–114.
113. ∫₁^(1/x) (1 / t) dt,x > 0

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Identify the integral to be evaluated: \(\int_{1}^{\frac{1}{x}} \frac{1}{t} \, dt\) where \(x > 0\).

Recall the antiderivative of the integrand \(\frac{1}{t}\), which is \(\ln|t|\).

Apply the Fundamental Theorem of Calculus, which states that \(\int_{a}^{b} f(t) \, dt = F(b) - F(a)\), where \(F\) is an antiderivative of \(f\).

Substitute the limits of integration into the antiderivative: \(\ln\left| \frac{1}{x} \right| - \ln|1|\).

Simplify the expression using logarithm properties, such as \(\ln\left( \frac{1}{x} \right) = -\ln(x)\) and \(\ln(1) = 0\), to express the integral in terms of \(x\).

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주요 개념

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Definite Integral and Variable Limits

A definite integral calculates the net area under a curve between two limits. When the limits themselves depend on a variable, the integral becomes a function of that variable, requiring careful evaluation of the integral with respect to those limits.

Integral of 1/t

The integral of 1/t with respect to t is the natural logarithm function, ln|t|, plus a constant. This fundamental integral is essential for solving problems involving logarithmic functions and appears frequently in calculus.

Properties of the Natural Logarithm

The natural logarithm has properties such as ln(a/b) = ln(a) - ln(b), which simplify expressions involving logarithms. Understanding these properties helps in manipulating and simplifying the results of integrals with variable limits.

Evaluate the integrals in Exercises 111–114.113. ∫₁^(1/x) (1 / t) dt,x > 0