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

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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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