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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?

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    i mean the first one2012-03-09

1 Answers 1

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Do you know the Riemann-Lebesgue lemma?

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    Follows quite trivi$a$lly from this lemma. Good jo$b$, +1!2012-03-09