sinx2 does not converge as x→∞, yet its integral from 0 to ∞ does.
I'm trying to understand why and would like some help in working towards a formal proof.
Answer
x↦sin(x2) is integrable on [0,1], so we have to show that lim exists. Make the substitution t=x^2, then x=\sqrt t and dx=\frac{dt}{2\sqrt t}. We have \int_1^A\sin(x^2)dx=\int_1^{A^2}\frac{\sin t}{2\sqrt t}dt=-\frac{\cos A^2}{2\sqrt A}+\frac{\cos 1}2+\frac 12\int_1^{A^2}\cos t\cdot t^{-3/2}\frac{-1}2dt,
and since \lim_{A\to +\infty}-\frac{\cos A^2}{2\sqrt A}+\frac{\cos 1}2=\frac{\cos 1}2 and the integral \int_1^{+\infty}t^{-3/2}dt exists (is finite), we conclude that \int_1^{+\infty}\sin(x^2)dx and so does \int_0^{+\infty}\sin(x^2)dx.
This integral is computable thanks to the residues theorem.
No comments:
Post a Comment