It seems to me that you lack basic knowledge about integration.
I suggest you download asap a calculus book and start to revise the chapter on integration.
you may go here : Pauls Online Math Notes
Now some hints...if u=x^2+2x+2 then du =(2x+2)dx therefore dx=(du)/(2x+2). du is the differential of the function u and as such it is equal to the derivative of u times the diferrential of x which is dx...
There is nothing wrong with the numerator....the first integral can be manipulated to become ( the numerator) d[(x+1)^2+1] and as such it will give the ln(x^2+2x+2) after integration. the second one is related to the inverse trigonometric functions...do you have any idea of what I am talking about? please revise the related topic.
Anyway the answer you post is correct.