Evaluate the integrals in Exercises 33–54.
∫(from ln3 to ln2) (e^x) dx

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Identify the integral to be evaluated: \(\int_{\ln 3}^{\ln 2} e^{x} \, dx\).

Recall the antiderivative of the integrand \(e^{x}\), which is \(e^{x}\) itself.

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

Substitute the limits of integration into the antiderivative: calculate \(e^{\ln 2} - e^{\ln 3}\).

Simplify the expressions using the property \(e^{\ln a} = a\) to rewrite the result in terms of numbers.

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주요 개념

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Definite Integral

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Antiderivative of the Exponential Function

The antiderivative of e^x is e^x itself, since the derivative of e^x is e^x. This property simplifies integration of exponential functions, allowing direct substitution of the limits after integration.

Properties of the Natural Logarithm

The natural logarithm ln(x) is the inverse of the exponential function e^x. Understanding that e^(ln a) = a helps simplify expressions when evaluating definite integrals with logarithmic limits, making it easier to compute exact values.

Evaluate the integrals in Exercises 33–54.∫(from ln3 to ln2) (e^x) dx