We have an answer to "Proving that an ∫−∞∞∫kk+1sin⁡(exp⁡(x))dxdk integral converges"

nemired9

Answered question

2022-01-14

Proving that an

\int_{- \infty}^{\infty} \int_{k}^{k + 1} \sin (\exp (x)) d x d k

integral converges

ambarakaq8

Beginner2022-01-15Added 31 answers

I = \int_{- \infty}^{+ \infty} \int_{e^{k}}^{e^{k + 1}} \frac{\sin z}{z} d z d k = \int_{- \infty}^{+ \infty} [S i (e^{k + 1}) - S i (e^{k})] d k = \int_{0}^{+ \infty} \frac{S i (e u) - S i (u)}{u} d u

and we may invoke Frullani's integral to get

I = \frac{π}{2}

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