How to prove $ \mathrm{e}^x\left|\int_x^{x+1}\sin\mathrm e^t \,\mathrm d t\right|\leqslant 2$?

Asked Mar 08 '20 at 16:35

Active Mar 08 '20 at 18:49

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I want to prove $$ \mathrm{e}^x\left|\int_x^{x+1}\sin\mathrm e^t \,\mathrm d t\right|\leqslant 2. $$ I tried to use Cauchy–Schwarz inequality and some other basic integral inequalities but failed.

Any suggestions will be appreciated.

edited Mar 08 '20 at 18:49

Michael Hardy

asked Mar 08 '20 at 16:35

bernard

The sharper bound $1.4$ also seems to hold but I don't know how to prove it – Maximilian Janisch Mar 08 '20 at 16:54
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@metamorphy Follow-up question – Maximilian Janisch Mar 08 '20 at 17:19

How to prove $ \mathrm{e}^x\left|\int_x^{x+1}\sin\mathrm e^t \,\mathrm d t\right|\leqslant 2$?

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