Solution:
Using DCT:
\( \frac{x^2}{\sqrt{x^{10}+5}} \) and \( \frac{1}{x^3} \) are ctn. on \( [5,\infty) \)
And \( 0 \leq \frac{x^2}{\sqrt{x^{10}+5}} \leq \frac{1}{x^3} \) on \( [5,\infty) \)
And \( \int \limits _5 ^\infty \frac{1}{x^3}dx \) converges \( p=3 >1 \)
Therefore, \( \int \limits _5 ^\infty \frac{x^2}{\sqrt{x^{10}+5}} \) converges by DCT
Using LCT:
\( \frac{x^2}{\sqrt{x^{10}+5}} \) and \( \frac{1}{x^3} \) are ctn. and pos. \( [5,\infty) \)
And \( \lim \limits _{x \rightarrow \infty} \frac{\frac{x^2}{\sqrt{x^{10}+5}}}{\frac{1}{x^3}} = \lim \limits _{x \rightarrow \infty}
\frac{x^5}{\sqrt{x^{10}+5}}=\lim \limits _{x \rightarrow \infty}
\frac{1}{\sqrt{1+\frac{5}{x^{10}}}}=1 \)
Since \( 0 < 1 < \infty \), and \( \int \limits _5 ^\infty \frac{1}{x^3}dx \) converges \( p=3 >1 \)
\( \int \limits _5 ^\infty \frac{x^2}{\sqrt{x^{10}+5}} \) converges by LCT
