what is the value of this integral ? $$\int_0^\infty \frac{\cos(\log(x))}{1+x^\pi}\sin(x)\,dx=\text{?}$$
we have $$\cos(\log(x))=\sum_{n=0}^\infty \frac{(-1)^n\log(x)^{2n}}{2n!}$$ 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 $\cos(x)$ .