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Aus Online Mathematik Brückenkurs 2

Observe that the derivative of the denominator is, for the most part, equal to the numerator,

x2+2x+2=2x+2=2x+1 

so we can rewrite the integral as

21x2+2x+2x2+2x+2dx 

The substitution u=x2+2x+2 will therefore simplify the integral considerably:

x+1x2+2x+2dx=u=x2+2x+2du=x2+2x+2dx=2x+1dx=21udu=21lnu+C=21lnx2+2x+2+C

NOTE: By completing the square

x2+2x+2=x+12−12+2=x+12+1 

we see that x2+2x+2 is always greater than or equal to 1, so we can take away the absolute sign around the argument in ln and answer with

21lnx2+2x+2+C