Example 5.5.6: The improper integral co sin(x2) (x+2) dx x+1 converges. Proof: Observe we simply need to show that the integral converges when going from 1 to infinity. For x > I we obtain sin (x2) (x+ 2) x+2 x+2 x +2x S x3+1 x3 + 1 S S S Then 3 dx = 3 x2 dx = 3. So using the comparison test and the tail test, the original integral converges