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 -1 to 1 of (dθ / (θ² - 2θ))

Accepted Answer

First, identify the integrand and the interval of integration. The integral is $$ \int_{-1}^{1} \frac{d	heta}{	heta^{2} - 2	heta} . $$Notice that the denominator can be factored as $$ 	heta(	heta - 2) . $$Check for points where the integrand is undefined or discontinuous within the interval $$[-1, 1]. $$Since the denominator is zero when $$ 	heta = 0  or$$ $$ 	heta = 2 $$, and $$ 0 $$ lies within the interval, the integral is improper at $$ 	heta = 0 . $$Split the integral at the point of discontinuity to handle the improper integral properly: $$ \int_{-1}^{1} \frac{d	heta}{	heta(	heta - 2)} = \int_{-1}^{0} \frac{d	heta}{	heta(	heta - 2)} + \int_{0}^{1} \frac{d	heta}{	heta(	heta - 2)} . To $$test for convergence, analyze the behavior of the integrand near $$ 	heta = 0 . $$Compare $$ \frac{1}{	heta(	heta - 2)} $$ near zero to a simpler function that dominates or is dominated by it, such as $$ \frac{1}{	heta} $$, since near zero $$ 	heta - 2  is $$approximately $$-2$$, a constant. Use the Direct Comparison Test or Limit Comparison Test with the simpler function $$ \frac{1}{|	heta|}  on $$each side of zero to determine if the improper integral converges or diverges near $$ 	heta = 0 .$$

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 -1 to 1 of (dθ / (θ² - 2θ))

Verified video answer for a similar problem:

Key Concepts

Improper Integrals and Convergence

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 -1 to 1 of (dθ / (θ² - 2θ))

Verified video answer for a similar problem:

Key Concepts

Improper Integrals and Convergence

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 -1 to 1 of (dθ / (θ² - 2θ))