<s>[INST] Detailing two integration equalities Could you please detail the follwing two equalities (I can not understand why they are valid):
\begin{align*}
\int_{0}^{\infty}\frac{\sin(2 \pi x)}{x+1}\, dx&=\int_{0}^{\infty}\frac{\sin( \pi x)}{x+2}\,dx\\
&=\int_{0}^{1}\sin( \pi x) \left[ \frac{1} {x+2} -\frac{1} {x+3}+\frac{1} {x+4}-\frac{1} {x+5}+\dotsb\right] \,dx
\end{align*}
[EDIT] the first one is a simple variable change. But the second I did not get.
 [/INST] The second one in the OP is not quite correct.  The argument of the sine function should be $\pi x $, not $2\pi x$. 
Note that we can write
$$\begin{align}
\int_0^\infty \frac{\sin(\pi x)}{x+2}\,dx&=\sum_{k=1}^\infty \int_{k-1}^k \frac{\sin(\pi x)}{x+2}\,dx\\\\
&=\sum_{k=1}^\infty \int_0^1 \frac{\sin(\pi (x+k-1))}{x+k+1}\,dx\\\\
&=\sum_{k=1}^\infty \int_0^1 \frac{(-1)^{k-1}\sin(\pi x)}{x+k+1}\,dx\\\\
&=\int_0^1 \sin(\pi x)\left(\frac{1}{x+2}-\frac{1}{x-3}+\frac{1}{x+4}-\frac{1}{x+5}+-...\right)\,dx
\end{align}$$
</s>