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 $\cos (ix)$

So Initially what i did was this

**
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.

The answer is 1.

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While it is true that $e^{i\times ix} = \cos(ix) + i\sin(ix),$ we should note that $\sin(ix)$ is not real, so the real part of this expression is not simply $\cos(ix)$ .

Rather, since $\cos(ix)$ is real and $\sin(ix)$ is purely imaginary [under the assumption $x$ is real], we see that $\Re\left(e^{i\times ix}\right) = \cos(ix) + i\sin(ix).$

Therefore, Step $\boxed{1}$ has the error.