1. ## Convergence of sin(x^2)

Hello,

I am reading a calculus book and I am stuck with the following exercise:

Show if $\lim_{t\rightarrow\infty}\int_1^t \sin(x^2) dx$ is convergent or not.

Sadly, the solutions at the end of the book only say that it is convergent but don't give any hint on how to show this. I would appreciate any ideas as I have spent quite some time on it already and it nags me a bit. Thanks in advance.

2. yes, it converges.

the key is to use an integration by parts.

first note that $\int_{1}^{\infty }{\frac{1-\cos x^{2}}{x^{2}}\,dx}<\infty,$ this is easy to verify since the integrand is positive then direct comparison test applies and $\frac{1-\cos x^{2}}{x^{2}}\le \frac{2}{x^{2}}.$

as for the problem, as i said, integrate by parts and get:

$\int_{1}^{\infty }{\sin \left( x^{2} \right)\,dx}=\frac{1}{2}\int_{1}^{\infty }{\frac{\left( 1-\cos x^{2} \right)'}{x}\,dx}=\frac{1}{2}\left( \underbrace{\left. \frac{1-\cos x^{2}}{x} \right|_{1}^{\infty }}_{\text{finite}}+\underbrace{\int_{1}^{\infty }{\frac{1-\cos x^{2}}{x^{2}}\,dx}}_{<\,\infty } \right).$

and we're done!

3. Thank you!