how to solve this $\lim_{a \to \infty}$ $\int_1^2 {\sin (ax)}/x^{2}(x-1)^{1/2}dx $ I want to compute
$$\lim_{t \to \infty} \int_1^2 \frac{\sin (tx)}{x^{2}(x-1)^{1/2}}\,dx.
$$ 
The integrand has discontinuity at $x=1$, so the integral is equal to the following limit:
$$\lim_{t \to \infty}\lim_{s \to 1^+} \int_s^2 \frac{\sin (tx)}{x^{2}(x-1)^{1/2}}dx, $$ 
and I use substitution $tx= a$; then $tdx=da$.
$$\lim_{t \to \infty}t^{3/2}\lim_{s \to 1^+}\int_{st}^{2t} \frac{\sin (a)}{a^{2}(a-t)^{1/2}}da $$
how to proceed this integral?

Do you know the Riemann-Lebesgue lemma?
