Evaluate the integrals in Exercises 33–54.
∫₀^(π/4) (1 + e^(tan θ)) sec²θ dθ

검증된 단계별 안내

Identify the integral to be evaluated: \(\int_0^{\frac{\pi}{4}} (1 + e^{\tan \theta}) \sec^2 \theta \, d\theta\).

Recognize that the integrand contains \(\sec^2 \theta\) and \(e^{\tan \theta}\), suggesting a substitution involving \(\tan \theta\) because the derivative of \(\tan \theta\) is \(\sec^2 \theta\).

Let \(u = \tan \theta\). Then, compute \(du = \sec^2 \theta \, d\theta\), which means \(\sec^2 \theta \, d\theta = du\).

Change the limits of integration from \(\theta\) to \(u\): when \(\theta = 0\), \(u = \tan 0 = 0\); when \(\theta = \frac{\pi}{4}\), \(u = \tan \frac{\pi}{4} = 1\).

Rewrite the integral in terms of \(u\): \(\int_0^1 (1 + e^u) \, du\). Then, split the integral into two simpler integrals: \(\int_0^1 1 \, du + \int_0^1 e^u \, du\).

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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 function and its derivative.

Derivative of the Tangent Function

The derivative of tan(θ) with respect to θ is sec²(θ). Recognizing this relationship helps in substitution because if the integrand contains sec²(θ) dθ, it can be replaced by d(tan θ), simplifying the integral.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two limits. When performing substitution, the limits of integration must be adjusted to correspond to the new variable, ensuring the integral is evaluated correctly within the new bounds.

Evaluate the integrals in Exercises 33–54.∫₀^(π/4) (1 + e^(tan θ)) sec²θ dθ