lim_(n→∞) ∫_0 ^1 e^x^2 sin(nx)dx


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Question Number 174883 by infinityaction last updated on 13/Aug/22

$\lim_{n \to \infty} \int_{0}^{1} e^{x^{2}} \sin (n x) d x$
	Answered by Mathspace last updated on 14/Aug/22

	$u_{n} = \int_{0}^{1} e^{x^{2}} s i n (n x) d x$ $\Rightarrow u_{n} =_{n x = t} \int_{0}^{n} e^{\frac{t^{2}}{n^{2}}} s i n t \frac{d t}{n}$ $= \int_{R} \frac{1}{n} e^{\frac{t^{2}}{n^{2}}} s i n t χ_{[0, n [} (t) d t$ $= \int_{R} f_{n} (t) d t$ $f_{n} \to 0 (c s) \Rightarrow l i m u_{n} = 0$
	Answered by Mathspace last updated on 14/Aug/22

	$a n o t h e r w a y$ $b y ρ a r t s$ $u_{n} = {[- \frac{1}{n} c o s (n x) e^{x^{2}}]}_{0}^{1}$ $+ \frac{1}{n} \int_{0}^{1} e^{x^{2}} c o s (n x) d x$ $\frac{1}{n} - e \frac{c o s n}{n} + \frac{1}{n} \int_{0}^{1} e^{x^{2}} c o s (n x) d x$ $∣ u_{n} ∣⩽ \frac{1}{n} + \frac{e}{n} + \frac{1}{n} \int_{0}^{1} e^{x^{2}} d x \to 0 (n \to + \infty)$ $\Rightarrow l i m u_{n} = 0$
	Commented by infinityaction last updated on 14/Aug/22

	$t h a n k y o u s i r$
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