I did the second step for this question but I can not fix it, and How can I fix this integral
2026-04-01 01:04:11.1775005451
Finding density of $X$ when it has characteristic function $\varphi(t)=(1+|t|)e^{-|t|}$
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Because $\phi(t)$ is an even function of $t$ $f_x(x)=\frac{1}{\pi}\int_0^\infty cos(tx)(1+t)e^{-t}dt=\frac{1}{\pi}(\frac{1}{1+x^2}+\frac{1-x^2}{(1+x^2)^2})=\frac{2}{\pi(1+x^2)^2}$