009B Sample Final 3, Problem 7

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{1}^{\infty }{\frac {\sin ^{2}(x)}{x^{3}}}~dx}

Foundations:
Direct Comparison Test for Improper Integrals
Let  $f$  and  $g$  be continuous on  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [a,\infty )}
where  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0\leq f(x)\leq g(x)}   for all  $x$  in  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [a,\infty ).}
1.  If  $\int _{a}^{\infty }g(x)~dx$  converges, then  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{a}^{\infty }f(x)~dx}   converges.
2.  If  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{a}^{\infty }f(x)~dx}   diverges, then  $\int _{a}^{\infty }g(x)~dx$  diverges.

Solution:

Step 1:
We use the Direct Comparison Test for Improper Integrals.
For all  $x$  in  $[1,\infty ),$
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0\leq {\frac {\sin ^{2}(x)}{x^{3}}}\leq {\frac {1}{x^{3}}}.}
Also,
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {\sin ^{2}(x)}{x^{3}}}}   and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{x^{3}}}}
are continuous on  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [1,\infty ).}

Step 2:
Now, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {\int _{1}^{\infty }{\frac {1}{x^{3}}}~dx}&=&\displaystyle {\lim _{a\rightarrow \infty }\int _{1}^{a}{\frac {1}{x^{3}}}~dx}\\&&\\&=&\displaystyle {\lim _{a\rightarrow \infty }{\frac {1}{-2x^{2}}}{\bigg |}_{1}^{a}}\\&&\\&=&\displaystyle {\lim _{a\rightarrow \infty }{\frac {1}{-2a^{2}}}+{\frac {1}{2}}}\\&&\\&=&\displaystyle {{\frac {1}{2}}.}\end{array}}}
Since  $\int _{1}^{\infty }{\frac {1}{x^{3}}}~dx$  converges,
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_1^\infty \frac{\sin^2(x)}{x^3}~dx}
converges by the Direct Comparison Test for Improper Integrals.

Final Answer:
converges (by the Direct Comparison Test for Improper Integrals)

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