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Let $$\frac{d}{dx}F(x)=\frac{e^{\sin x}}{x},\quad\,x>0$$ if $$\int_1^4\frac{2e^{\sin x^2}}{x}\ dx=F(k)-(1)$$

then one of the possible value of $k$ is?

I guess we could some $u$ substitution for the equation on row $2$, but I'm not entirely sure how, and would appreciate some insights. The answer is $16$.

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  • $\begingroup$ Is it $\sin(x^2)$ or $\sin^2(x)$ ? $\endgroup$
    – Claude Leibovici
    Commented Sep 16, 2023 at 6:43
  • $\begingroup$ @ JiaoCtagon. I did answer your question? $\endgroup$
    – bob
    Commented Sep 17, 2023 at 9:06
  • $\begingroup$ @JiaoCtagon. I see you’ve set a bounty and are looking for a “reputable source”. Tbh, I don’t see how an answer from a reputable source will change anything for a question like this. If you don’t understand my hints, take a look at the spoiler. If you still don’t understand with the spoilers, you can ask in the comments. $\endgroup$
    – bob
    Commented Sep 18, 2023 at 10:07

1 Answer 1

Reset to default
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Hint: Let $x=\sqrt t$ and then use the definition of a definite integral.

Edit: I don't really know why I have a down vote, but my hints do the job.

After $x=\sqrt t$,
$$\int_1^{16}\frac{e^{\sin t}}{t}\ dt$$ now just look at the upper bound.

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    $\begingroup$ nice hidden answer... I didn't know you could do that... it was the $>!$ characters? $\endgroup$
    – Joako
    Commented Sep 19, 2023 at 18:50
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    $\begingroup$ @Joako. It's pretty neat. I didn't know about it too until I saw someone else use it recently. $\endgroup$
    – bob
    Commented Sep 20, 2023 at 4:43

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