Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt

검증된 단계별 안내

Start by examining the integral \( \int \frac{e^{4t} + 2e^{2t} - e^{t}}{e^{2t} + 1} \, dt \). Notice that the numerator and denominator involve exponential functions with powers related to \( t \).

Make a substitution to simplify the expression. Let \( u = e^{2t} \). Then, compute \( du \) in terms of \( dt \): \( du = 2e^{2t} dt = 2u dt \), so \( dt = \frac{du}{2u} \).

Rewrite the integral in terms of \( u \). Express each term in the numerator and denominator using \( u \): \( e^{4t} = (e^{2t})^2 = u^2 \), \( 2e^{2t} = 2u \), and \( e^{t} = e^{t} \) (which can be expressed as \( e^{t} = (e^{2t})^{1/2} = u^{1/2} \)). Substitute these into the integral and replace \( dt \) with \( \frac{du}{2u} \).

Simplify the resulting integral in terms of \( u \), combining like terms and simplifying the fraction if possible. This should reduce the integral to a rational function of \( u \) and possibly \( u^{1/2} \).

Once simplified, split the integral into simpler parts if needed and integrate each term with respect to \( u \). After integrating, substitute back \( u = e^{2t} \) to express the answer in terms of \( t \).

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주요 개념

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

Integration of Rational Functions

This involves integrating expressions where one function is divided by another, often requiring algebraic manipulation such as polynomial division or substitution to simplify the integrand before integration.

Exponential Functions and Their Properties

Understanding the rules of exponents, such as e^{a} * e^{b} = e^{a+b}, and how to rewrite expressions with exponential terms is crucial for simplifying the integrand and identifying substitution opportunities.

Substitution Method in Integration

This technique involves changing variables to simplify the integral, often by letting a part of the integrand equal a new variable, which transforms the integral into a more manageable form.

Evaluate the integrals in Exercises 39–54.∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt