29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ e^{2x} / (4 + e^{2x}) dx

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Identify the integral to solve: \(\int \frac{e^{2x}}{4 + e^{2x}} \, dx\).

Use substitution to simplify the integral. Let \(u = e^{2x}\). Then, compute \(du\): since \(u = e^{2x}\), differentiate to get \(du = 2 e^{2x} dx\), or equivalently, \(dx = \frac{du}{2u}\).

Rewrite the integral in terms of \(u\): substitute \(e^{2x} = u\) and \(dx = \frac{du}{2u}\) into the integral to get \(\int \frac{u}{4 + u} \cdot \frac{du}{2u} = \int \frac{1}{4 + u} \cdot \frac{1}{2} du = \frac{1}{2} \int \frac{1}{4 + u} du\).

Integrate \(\frac{1}{4 + u}\) with respect to \(u\): recall that \(\int \frac{1}{a + u} du = \ln|a + u| + C\), so here the integral becomes \(\frac{1}{2} \ln|4 + u| + C\).

Substitute back \(u = e^{2x}\) to express the answer in terms of \(x\): the integral is \(\frac{1}{2} \ln|4 + e^{2x}| + C\).

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주요 개념

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

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function.

Exponential Functions

Exponential functions have the form e^{kx}, where e is Euler's number and k is a constant. Their derivatives and integrals often involve the same exponential function, making them straightforward to handle in calculus. Recognizing these functions helps in applying substitution effectively.

Handling Absolute Values in Integrals

Absolute values appear in integrals when dealing with logarithmic functions resulting from integration. They ensure the argument of the logarithm is positive, maintaining the function's domain. Including absolute values only when necessary avoids incorrect expressions and ensures the integral's correctness.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.∫ e^{2x} / (4 + e^{2x}) dx