Question

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.84. ∫ (from 0 to π) sec²x dx*(Note: Potential improperness at x = π/2)*

Accepted Answer

Identify the integral to evaluate: $$\$$int_0$$^{\pi} \sec^2$ x \, dx. $$Notice that $$\sec^2$ x = \frac{1}{\cos^2$ x}$$, and the function has a potential vertical asymptote where $$\cos x = 0$$, which occurs at $$x = \frac{\pi}{2}$$ within the interval $$[0, \pi]. $$Since the integrand is not defined at $$x = \frac{\pi}{2}$$, split the integral into two improper integrals at this point: $$\$$int_0$$^{\pi} \sec^2$ x \, dx = \$$int_0$$^{\frac{\pi}{2}} \sec^2$ x \, dx + \int_{\frac{\pi}{2}}^{\pi} \sec^2$ x \, dx. $$Rewrite each integral as a limit to handle the improper behavior: $$\$$int_0$$^{\frac{\pi}{2}} \sec^2$ x \, dx = \lim_{t 	o \frac{\pi}{2}^-} \$$int_0$$^t \sec^2$ x \, dx$$ and $$\int_{\frac{\pi}{2}}^{\pi} \sec^2$ x \, dx = \lim_{s 	o \frac{\pi}{2}^+} \int_s$$^{\pi}$$ \sec^2$ x \, dx. $$Find the antiderivative of $$\sec^2$ x$$, which is $$	an x$$, and express each integral in terms of $$	an x$$: $$\int \sec^2$ x \, dx = 	an x + C. $$Evaluate the limits for each integral using the antiderivative: compute $$\lim_{t 	o \frac{\pi}{2}^-} (	an t - 	an 0)$$ and $$\lim_{s 	o \frac{\pi}{2}^+} (	an \pi - 	an s)$$, then analyze whether these limits converge or diverge to determine the behavior of the original integral.

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.
84. ∫ (from 0 to π) sec²x dx(Note: Potential improperness at x = π/2)

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Concetti chiave

Improper Integrals

Behavior of sec²x and its Discontinuities

Evaluating Limits in Definite Integrals