Calculus Level pending

A problem was given by Akeel Howell , Here you get it Complex Limit

Here he says that to find $\large \lim_{t \to \infty}{\int_{0}^{\ln{t}}{\cos{(ix)}} \, dx} =\ \dfrac{a}{b}t$

Now 1st of all we need $$\cos (ix)$$

Step 1 $$\cos(ix)$$ = $$Re(e^{i \times (ix)})$$ By Euler's Method

Step 2 So I got $$Re(e^{-x})$$

Step 3 Then as $$e^{-x}$$ is Fully real so $$Re(e^{-x}) = e^{-x} = \cos(ix)$$

Step 4 But we know $$\cos (ix) = coshx = \dfrac{e^x+e^{-x}}{2}$$

So whats wrong in evaluating $$\cos (ix)$$?

Input your answer as the step number. If you think whole thing is wrong input $$10$$ but give reason.

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