What is the valus of this integral?

Mathematics Asked by Bachamohamed on January 5, 2022

what is the value of this integral ?
$$int_0^infty frac{cos(log(x))}{1+x^pi}sin(x),dx=text{?}$$

we have $$cos(log(x))=sum_{n=0}^infty frac{(-1)^nlog(x)^{2n}}{2n!}$$ And from it we find $$int_0^infty frac{cos(log(x))}{1+x^{pi}} = sum_{n=0}^infty frac{(-1)^{n}}{(2n)!}int_0^infty frac{log(x)^{2n}}{1+x^pi}sin(x) , dx$$ we will arrive at another integration that needs to be calculated more than the first integration. Please help and give an opinion on this account Even with the added use, the cauchy is useful for the two series $sin(x)$ and $cos(x)$ .

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