Both methods are correct. Your mistake is in the third line, you just made a mistake in your integration.
With this correction, your other 2x vanishes and you are left with a similar expression.
There's nothing wrong with your solutions.
Also it's not necessary to use 'absolute value signs' because e^(-x)+1>0 for each x.
That's not a mistake because:
So just the same as Gralla55 said.