Finding $\int_{0}^{\infty}\frac{e^{-t}\cos(t)}{t}\text{ d}t$

Question

I know that $$\int_{0}^{\infty}\frac{f(t)}{t}=\int_{0}^{\infty}\mathcal L \{ f(t) \} \text{ d}s$$

So

$$\int_{0}^{\infty}\frac{e^{-t}\cos(t)}{t}\text{ d}t=\int_{0}^{\infty}\mathcal L \{e^{-t}\cos(t)\} \text{ d}s$$

$$=\int_{0}^{\infty}\frac{s+1}{(s+1)^2+1}\text{ d}s$$

$$=\frac{1}{2}\tan^{-1}(s+1)|_{0}^{\infty}$$

$$=\frac{1}{2}\bigg(\frac{\pi}{2}-\frac{\pi}{4}\bigg)=\frac{\pi}{8}$$

• Calculating the Laplace transform was obtained using WA.

• The integral does not converge as WA says.

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Where did I make a mistake? I know that this function is not continuous as the denominator goes to $0$, but that is not a reason for an integral to fail to converge.

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Your help would be appreciated. THANKS!