- $\int_0^\infty \frac{\cos (x)}{1+4x^2}\, dx$
- $\int_0^\infty \frac{1}{(1+x^2)^2}\, dx$
There is no hint for these two questions. I think for Q2, since it's a square, I can use Plancherel formula for $e^{-2\pi|x|}$. But I am not sure how to solve the first one.
Let $f(t) = e^{-|t|}$, then with $\hat{f} = {\cal F} f$, we have $\hat{f}(x) ={2 \over 1+x^2}$. The inversion theorm gives
\begin{eqnarray} f(t) &=& {1 \over 2\pi}\int_{\mathbb{R}} {2 e^{i xt} \over 1 + x^2} dx\\ &=& {1 \over \pi}\int_{-\infty}^0 {e^{i xt} \over 1 + x^2} dx + {1 \over \pi}\int_0^\infty {e^{i xt} \over 1 + x^2} dx\\ &=& {1 \over \pi}\int_0^\infty {e^{i xt} + e^{-ixt}\over 1 + x^2} dx\\ &=& {2 \over \pi} \int_0^\infty {\cos(xt)\over 1 + x^2} dx \\ &=& {4 \over \pi} \int_0^\infty {\cos(2yt)\over 1 + (2y)^2} dy \\ \end{eqnarray} and so (1) is given by ${\pi \over 4} f({1 \over 2})$.
Look at $f(0)$ and the Plancherel/Parseval identity for (2).