29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₋₂² (e^{z/2}) / (e^{z/2} + 1) dz

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First, observe the integral \( \int_{-2}^{2} \frac{e^{z/2}}{e^{z/2} + 1} \, dz \). Notice the integrand involves an exponential function in the numerator and denominator.

To simplify the integral, consider the substitution \( u = e^{z/2} \). Then, \( du = \frac{1}{2} e^{z/2} dz = \frac{1}{2} u \, dz \), which implies \( dz = \frac{2}{u} du \). However, since the limits are in terms of \( z \), it might be easier to explore symmetry or another approach first.

Check if the integrand has any symmetry properties by evaluating \( f(-z) \) and comparing it to \( f(z) \). This can help determine if the integral can be simplified by splitting or combining parts.

Rewrite the integrand as \( \frac{e^{z/2}}{e^{z/2} + 1} = 1 - \frac{1}{e^{z/2} + 1} \). This decomposition can make the integral easier to handle by splitting it into two integrals.

Express the original integral as \( \int_{-2}^{2} 1 \, dz - \int_{-2}^{2} \frac{1}{e^{z/2} + 1} \, dz \). Then, evaluate each integral separately, using substitution or symmetry as appropriate.

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주요 개념

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

Definite Integrals and Symmetry

Definite integrals calculate the net area under a curve between two limits. When the integrand exhibits symmetry (even, odd, or other), this property can simplify evaluation by reducing the interval or transforming the integral.

Substitution Method

The substitution method involves changing variables to simplify an integral. By letting a part of the integrand equal a new variable, the integral can often be rewritten in a more manageable form, facilitating easier integration.

Exponential Functions and Their Properties

Exponential functions like e^{x} have unique properties, such as their derivatives and integrals being proportional to themselves. Understanding how to manipulate expressions involving exponentials is crucial for integrating functions with exponential terms.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.∫₋₂² (e^{z/2}) / (e^{z/2} + 1) dz