Counterintuitive integration

Notice that the integrand can be rewritten as

$\dfrac{(\cos x) \left( x^4 + 5x \right) - \left( 4x^3 + 5 \right)(\sin x)}{\left(x^4 + 5x\right)^2}$

Doesn't that look like the quotient rule? It IS the quotient rule.

Then we have

$\left. \dfrac{\sin x }{x^4 + 5x} \right|_{\pi/2}^{\infty}$

which gives us

$\dfrac{\text{A number between } -1 \text{ and } 1}{\infty} - \dfrac{1}{\left( \dfrac{\pi}{2}\right)^4 + 5 \left(\dfrac{\pi}{2} \right)} = - \dfrac{1}{\left( \dfrac{\pi}{2}\right)^4 + 5 \left(\dfrac{\pi}{2} \right)}$

and that gives $1 + 4 + 5 + 1 = \boxed{11}$