My problem is to simplify the following integral:

I have a solution, but can't understand it without explanations. The solution is following:

1)

2)

, where , ,

The second case is clear: we expand the exponential term to the 2nd order Taylor series around u=1, simplify the expression, change variable, set the lower integration linit from 0 to and get the complimentary error function. But the first case result is not clear at all. Please, help me to understand. Thank you.