The limits of a definite integral using Mean Value theorem

Question

Show that: $$ -\frac{1}{2} < \int_0^1 \frac {x^3 cos 5x}{2+x^2} dx <\frac{1}{2}. $$ I have started by proving that for any two integrable functions $f, g$ defined on $R[a,b]$ such that $f>g$ on $ [a,b]$ if at any $c\in\ [a,b]$ $f$ and $g$ are continuous and $f(c) > g(c)$ then $$ \int_a^b f(x) dx > \int_a^b g(x) dx .$$ I also tried to use the fact that $$ |cos5x| <1.$$ How do I proceed from here? Thank you for your help.