This exercise will show that $cos(x^2)$ is an example of a function that oscillates back and forth from $-1$ to $1$ as $x$ goes to infinity, yet its integral ${\displaystyle \underset{1}{\overset{}{}}}cos(x^2)$ still converges.

a$)$ Using an appropriate integration by parts, show that for any $N>1$ one has the following.

${\displaystyle \underset{1}{\overset{N}{}}}\frac{sin(x^2)}{2x^2}=-\frac{sin(N^2)}{2N}+\frac{sin1}{2}+{\displaystyle \underset{1}{\overset{N}{}}}cos(x^2)$

b$)$ Use part a$)$ to show that the improper integral ${\displaystyle \underset{1}{\overset{}{}}}cos(x^2)$ converges.