Science:Math Exam Resources/Courses/MATH101 A/April 2024/Question 09/Hint 1

For a more elementary solution that doesn't use the limit comparison test, consider the following rewriting of the integrand:

$$\frac{x^2+1}{x^3+2}=\frac{x^2}{x^3+2}+\frac{1}{x^3+2}=\frac{1}{x+\frac{2}{x^2}}+\frac{1}{x^3+2}.$$ Can you use this to show that the integrand is bounded by $\frac{K}{x}$ for some $K$?

Alternatively, the second solution applies the limit comparison test.