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Aus Online Mathematik Brückenkurs 2

What makes our integral differ from that in the exercise´s text is that there is a term x2+4 instead of x2+1, but if we factor out the 4 from the denominator,

dxx2+4=dx441x2+1=41dx41x2+1,

we obtain the correct second term in the denominator. On the other hand, there is no longer x2 but 41x2, although we can get around this by substituting u=21x,

41dx41x2+1=41dx(x2)2+1=udu=x2=21dx=412duu2+1=21duu2+1=21arctanu+C=21arctanx2+C.