Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$.
Use mean value theorem $$\int_x^{x+1}\sin(e^t)dt=\sin(e^\xi)$$
And we have $$|\sin(e^\xi)|\le\frac{2}{e^x}$$
where $\xi\in(x,x+1)$
I stuck here. Both new methods and help me to continue are welcome.
2026-03-28 20:54:05.1774731245
Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$
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