Improper integral converges or diverges

Question

∫ (from 1 to ∞) (2+sin(x))/(x²+3x+5) dx

Answer 1

The best way to approach this is by using a comparison test.

Since −1≤sin⁡x≤1.

Also,

x2+3x+5≥x2.

Thus,

0≤2+sin⁡xx2+3x+5≤3x2.

Now,

∫1∞3x2 dx=3∫1∞x−2 dx=3,

By comparison test, the integral converges