Integrate $ I(t)=\int_0^\infty \left( \frac{\sin tx}{x} \right)^n\,\mathrm d x$

Asked May 29 '19 at 13:49

Active May 29 '19 at 14:16

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How to integrate the following: $$ I(t)=\int_0^\infty \left( \frac{\sin tx}{x} \right)^n\,\mathrm d x$$

I tried to using the Laplace transform:

\begin{align} \mathcal{L}\left[I(t)\right]=\int_0^\infty \mathcal L\left[\left( \frac{\sin tx}{x} \right)^n\right]\,\mathrm d x \end{align}

but I don't know what to do then.

edited May 29 '19 at 14:16

StubbornAtom

asked May 29 '19 at 13:49

糯米非米

3

First of make a change of variables $u = tx$ to get $I(t) = Ct^{n-1}$ where $C = \int_0^\infty \left(\frac{\sin(u)}{u}\right)^n{\rm d}u$. You now know the full $t$ dependence of the function and are left with having to evaluate the constant $C$. – Winther May 29 '19 at 14:16
And the the solution is found here https://math.stackexchange.com/q/307510/647013 – Zacky May 29 '19 at 18:05

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