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

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

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

$\frac{1}{8}\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x}dx=\frac{1}{8}\underset{\alpha \to \mathrm{\infty }}{\lim }[\text{ln}(x){]}_1^{\alpha }=\frac{1}{8}\underset{\alpha \to \mathrm{\infty }}{\lim }\text{ln}(\alpha )=\mathrm{\infty }$