Determine whether the improper integral converges or diverges.
-$$\int _ { 0 } ^ { \pi / 2 } \sec \theta d \theta$$
A) Diverges
B) Converges

Question

Quizplus · Accepted Answer

We can use the comparison test to show that the given integral diverges. Notice that for $0 \le 	heta < \frac{\pi}{2}$, we have $\cos 	heta \ge \frac{2}{\pi} 	heta$, which implies $\sec 	heta \ge \frac{\pi}{2} \csc 	heta$. Therefore,
$$\int_0^{\pi/2} \sec 	heta d	heta \ge \frac{\pi}{2} \int_0^{\pi/2} \csc 	heta d	heta$$
The integral on the right-hand side is a well-known divergent integral, so the integral on the left-hand side must also diverge.

Determine Whether the Improper Integral Converges or Diverges ∫0π/2sec⁡θdθ\int _ { 0 } ^ { \pi / 2 } \sec \theta d \theta∫0π/2​secθdθ

Determine Whether the Improper Integral Converges or Diverges $\int _ { 0 } ^ { \pi / 2 } \sec \theta d \theta$