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 (-x ln|x| dx)

Accepted Answer

First, recognize that the integral is improper because the integrand involves $$ \ln|x| $$, which is undefined at $$ x = 0 . So$$, we need to consider the integral as an improper integral with a potential discontinuity at $$ x = 0 . $$Rewrite the integral as the sum of two integrals around the point of discontinuity: $$ \int_{-1}^{1} -x \ln|x| \, dx = \int_{-1}^{0} -x \ln|x| \, dx + \int_{0}^{1} -x \ln|x| \, dx . $$Since the function $$ f(x) = -x \ln|x|  is an $$even function (because $$ -x \ln|x|  is $$symmetric about zero), you can simplify the problem by evaluating $$ 2 \int_{0}^{1} -x \ln x \, dx $$ instead. To test for convergence, analyze the behavior of the integrand near $$ x = 0 . $$Consider the limit $$ \lim_{x 	o 0^{+}} -x \ln x . $$Use L'Hôpital's Rule or known limits to determine if the integrand approaches a finite value or zero. Since the integrand behaves well near zero and is continuous on $$ (0,1] $$, conclude that the integral converges. You can also use direct integration techniques or comparison tests with functions like $$ x^{p}  to $$confirm convergence.

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 (-x ln|x| dx)

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Concetti chiave

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 (-x ln|x| dx)

Risposta video verificata per un problema simile:

Concetti chiave

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 (-x ln|x| dx)