Answer: Writing (a) The improper integrals and diverge and converge, respectively
Chapter 8, Problem 91(choose chapter or problem)
Writing
(a) The improper integrals
\(\int_{1}^{\infty} \frac{1}{x} d x \text { and } \int_{1}^{\infty} \frac{1}{x^{2}} d x\)
diverge and converge, respectively. Describe the essential differences between the integrands that cause one integral to converge and the other to diverge.
(b) Sketch a graph of the function y =(sin x)/x over the interval (1, \(\infty\)). Use your knowledge of the definite integral to make an inference as to whether the integral
\(\int_{1}^{\infty} \frac{\sin x}{x} d x\)
converges. Give reasons for your answer.
(c) Use one iteration of integration by parts on the integral in part (b) to determine its divergence or convergence.
Text Transcription:
Int_1^infinity 1/x dx and Int_1^infinity 1/x^2 dx
Infinity
Int_1^infinity sin x/x dx
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