7–58. Improper integrals Evaluate the following integrals or state that they diverge.
39. ∫ (from 0 to π/2) tan θ dθ

Verified step by step guidance

1. Identify the integral to evaluate: \(\int_0^{\frac{\pi}{2}} \tan \theta \, d\theta\).
2. Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), and the integral of \(\tan \theta\) can be expressed as \(-\ln |\cos \theta| + C\).
3. Check the behavior of the integrand at the limits of integration, especially at \(\theta = \frac{\pi}{2}\), where \(\cos \theta = 0\) and \(\tan \theta\) tends to infinity, indicating an improper integral.
4. Rewrite the integral as a limit to handle the improper behavior: \(\lim_{b \to \frac{\pi}{2}^-} \int_0^b \tan \theta \, d\theta\).
5. Evaluate the integral using the antiderivative \(-\ln |\cos \theta|\) and then take the limit as \(b\) approaches \(\frac{\pi}{2}\) from the left to determine if the integral converges or diverges.

Key Concepts

Improper Integrals

Improper integrals involve integrals with infinite limits or integrands that become unbounded within the interval. To evaluate them, one often takes limits approaching the problematic points to determine convergence or divergence.

Behavior of the Tangent Function near π/2

The tangent function, tan(θ), approaches infinity as θ approaches π/2 from the left. This vertical asymptote means the integral from 0 to π/2 is improper and requires careful limit evaluation to check if the area under the curve is finite.

Evaluating Limits in Definite Integrals

When an integral has an infinite discontinuity at an endpoint, it is evaluated as a limit. For example, ∫₀^{π/2} tan θ dθ is computed as the limit of ∫₀^{b} tan θ dθ as b approaches π/2 from below.