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  $[a,\infty ).$
1.  If  $\int _{a}^{\infty }g(x)~dx$  converges, then  $\int _{a}^{\infty }f(x)~dx$  converges.
2.  If  $\int _{a}^{\infty }f(x)~dx$  diverges, then  $\int _{a}^{\infty }g(x)~dx$  diverges.

Solution:

Step 1:

Step 2:

Final Answer:

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