Science:Math Exam Resources/Courses/MATH103/April 2012/Question 01 (b)/Solution 1

The difference between the highest exponents is 3 - 2 = 1. By the intergral p test (with p = 1) this integral diverges.

More precisely we estimate the integral with

$\int _{1}^{\infty }{\frac {x^{2}}{x^{3}+2x+5}}dx>\int _{1}^{\infty }{\frac {x^{2}}{x^{3}+2x^{3}+5x^{3}}}dx={\frac {1}{8}}\int _{1}^{\infty }{\frac {1}{x}}dx$

Here we see exactly how to use the integral p test. We also can continue with the anti-derivative

${\frac {1}{8}}\int _{1}^{\infty }{\frac {1}{x}}dx={\frac {1}{8}}\lim _{\alpha \rightarrow \infty }[{\text{ln}}(x)]_{1}^{\alpha }={\frac {1}{8}}\lim _{\alpha \rightarrow \infty }{\text{ln}}(\alpha )=\infty$