Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$

Asked Apr 05 '16 at 12:15

Active Apr 05 '16 at 13:10

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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.

edited Apr 05 '16 at 13:10

asked Apr 05 '16 at 12:15

Rowan

2

The integrand is highly oscillatory and the key is it mostly cancels out. So MVT estimate for the entire interval will be too rough. Better could be to estimate how much remains after complete sine waves are eliminated from the interval. – Macavity Apr 05 '16 at 12:56
Keywords: stationary phase principles, oscillatory integrals – Bananach Apr 05 '16 at 14:46

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