No.
The answer in the calculator is
Its not .
See this:
integrate [ e^[-x] ] / [ e^[-x] - 1 ] - Wolfram|Alpha
Check it please.
It is the same.
If you differentiate this answer and make a common denominator (here you will get a minus sign in the numerator of a fraction) and multiply the resulting fraction by and take the minus sign as common factor from the denominator and cancel it with the another minus sign, you will get the same function which we integrated it.
I know this maybe will confuse you.
Do the following steps:
1- Differentiate .
2- Make a common denominator to get a fraction with a minus sign in the numerator.
3- Multiply this fraction by .
4- take the minus sign as common factor from the denominator and cancel it with the another one.
5- GOT IT ?