Calculating $\int_0^\infty \frac{\cos(t)}{(1+t^2)^3}\text{d}t$ I am very new to the Residuetheorem and now I am asked to calculate the following integral:
$$\int_0^\infty \frac{\cos(t)}{(1+t^2)^3}\text{d}t$$
I know it has poles of order $3$ at $x=\pm i$ and that I have to find a closed curve in order to calculate it. 
But as I said, I am very new to this and (to be hounest) a little lost at the moment. Therefore any hint or help is very much appreciated!

Note that your integral is equal to$$\frac12\operatorname{Re}\left(\int_{-\infty}^\infty\frac{e^{it}}{(1+t^2)^3}\,\mathrm dt\right).$$And$$\int_{-\infty}^\infty\frac{e^{it}}{(1+t^2)^3}\,\mathrm dt=2\pi i\operatorname{res}_{z=i}\frac{e^{iz}}{(1+z^2)^3}.$$Finally, this last residue is equal to $-\frac{7i}{16e}$. Therefore, your integral is equal to $\frac{7\pi}{16e}$.
