Evaluate the integrals in Exercises 53–76.
73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))

검증된 단계별 안내

Recognize the integral to evaluate: \(\int_0^{\ln \sqrt{3}} \frac{e^x}{1 + e^{2x}} \, dx\).

Make a substitution to simplify the integral. Let \(u = e^x\). Then, the differential is \(du = e^x \, dx\), which means \(e^x \, dx = du\).

Rewrite the integral in terms of \(u\). The limits change as follows: when \(x = 0\), \(u = e^0 = 1\); when \(x = \ln \sqrt{3}\), \(u = e^{\ln \sqrt{3}} = \sqrt{3}\). The integral becomes \(\int_1^{\sqrt{3}} \frac{1}{1 + u^2} \, du\).

Recognize that \(\int \frac{1}{1 + u^2} \, du\) is the standard integral for \(\arctan u + C\). So, the integral evaluates to \(\arctan u\) evaluated from \(1\) to \(\sqrt{3}\).

Apply the Fundamental Theorem of Calculus by substituting the limits back into \(\arctan u\) to express the definite integral as \(\arctan(\sqrt{3}) - \arctan(1)\).

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주요 개념

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

Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the antiderivative at the upper and lower bounds and subtracting these values. This concept is essential for solving integrals with given limits, such as from 0 to ln(√3).

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It often involves setting a part of the integrand as a new variable, which helps in integrating complex expressions like those involving exponential functions.

Properties of Exponential Functions

Exponential functions, such as e^x, have unique properties including their derivatives and integrals being proportional to themselves. Understanding how to manipulate expressions like e^(2x) and their relationships is crucial for simplifying and evaluating integrals involving exponentials.

Evaluate the integrals in Exercises 53–76.73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))