Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to ∞ of (dθ / (θ² - 1))

Accepted Answer

First, identify the integral to be tested for convergence: $$\displaystyle \$$int_0$$^{\infty} \frac{d	heta}{\$$theta^2$$ - 1}. $$Notice that the integrand has vertical asymptotes where the denominator is zero, i.e., at $$	heta = 1. $$This means the integral is improper at $$	heta = 1$$ and at the upper limit $$\infty. $$Split the integral into two improper integrals at the point of discontinuity: $$\displaystyle \$$int_0$$^1 \frac{d	heta}{\$$theta^2$$ - 1} + \$$int_1$$^{\infty} \frac{d	heta}{\$$theta^2$$ - 1}. To $$test convergence near $$	heta = 1$$, analyze the behavior of the integrand as $$	heta 	o 1. $$For the limit at infinity, compare the integrand to a simpler function such as $$\frac{1}{\$$theta^2$$}$$ using the Direct Comparison Test or Limit Comparison Test. Apply the Direct Comparison Test or Limit Comparison Test by computing the limit $$\displaystyle \lim_{	heta 	o \infty} \frac{1/(\$$theta^2$$ - 1)}{1/\$$theta^2$$}$$ and check if the integral converges or diverges based on the known behavior of $$\$$int_1$$^{\infty} \frac{1}{\$$theta^2$$} d	heta.$$

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to ∞ of (dθ / (θ² - 1))

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Concetti chiave

Improper Integrals

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.∫ from 0 to ∞ of (dθ / (θ² - 1))

Risposta video verificata per un problema simile:

Concetti chiave

Improper Integrals

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to ∞ of (dθ / (θ² - 1))