how to evaluate lim_{n->infinity} int_0^1 e^{x^2} sin(nx) dx ?

[Density]

 

[stapel]

11. The value of [imath]\displaystyle \lim_{n \rarr \infty} \int_0^1 e^{x^2}\, \sin(nx)\, dx[/math] is ______? [/QUOTE] What are your thoughts? What have you tried? How far have you gotten? ("[URL='https://www.freemathhelp.com/forum/threads/read-before-posting.41538/']Read Before Posting[/URL]") Please be complete. Thank you![/imath]

 

[fresh_42]

Can you use the series expansion of [imath] \sin(nx) [/imath]?

 

[Density]

firstly i tried with the gamma function but it is not getting anywhere with that

 

[blamocur]

I'd try using [math]\int_0^1 e^{x^2} \sin (nx) dx = O\left(\frac{1}{n}\right) + \sum_k \int_{\frac{2\pi k}{n}}^{\frac{2\pi(k+1)}{n}} e^{x^2} \sin nx\;dx[/math] then replace [imath]u=nx[/imath] and use the fact that [imath]e^{x^2}[/imath] has limited derivativet to show that each summand is of order [imath]O\left(\frac{1}{n^2}\right)[/imath]