009B Sample Final 3, Problem 7

Does the following integral converge or diverge? Prove your answer!

$\int _{1}^{\infty }{\frac {\sin ^{2}(x)}{x^{3}}}~dx$

Foundations:
Direct Comparison Test for Improper Integrals
Let  $f$  and  $g$  be continuous on  $[a,\infty )$
where  $0\leq f(x)\leq g(x)$  for all  $x$  in  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,\infty).}
1.  If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty g(x)~dx}   converges, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty f(x)~dx}   converges.
2.  If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty f(x)~dx}   diverges, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^\infty g(x)~dx}   diverges.

Solution:

Step 1:

Step 2:

Final Answer:

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