#int (d x)/(sqrt (x^2+4))=?#
#"let be "u=x/2" ; " d x=2*d u" ; "x^2=4u^2#
#int (2*d u)/(sqrt(4u^2+4))=int(2*d u)/(sqrt(4(u^2+1))#
#int(cancel(2)* d u)/(cancel(2)*sqrt(u^2+1))=int(d u)/(sqrt(u^2+1))#
#"now, substitute "u=tan v" ; "v=arc tan u#
#d u=sec ^2 v* d v#
#int (sec ^2 v*d v)/(sqrt(tan^2 v+1))" ;so "tan^2 v +1=sec ^2 v#
#int(sec^2 v*d v)/(sqrt(sec^2 v))=int (cancel(sec)^2 v*d v)/(cancel(sec) v)=int sec v*d v#
#"expand fraction by " tan v+sec v#
#int sec v*d v*(tan v+sec v)/(tan v+sec v)#
#int (sec v*tan v + sec ^2 v)/(tan v + sec v) *d v#
#k=tan v+ sec v#
#d k=(sec v* tan v+sec ^2 v) *d v#
#int ( d k)/k=l n k+C#
#"undo substitution "k=tan v +sec v#
#int (d x)/(sqrt (x^2+4))= l n(tan v+sec v)+C#
#sec v=sqrt(1+tan ^2 v)=sqrt (1+u^2)#
#int (d x)/(sqrt (x^2+4))=l n(u+sqrt(1+u^2))+C#
#"undo substitution " u=x/2#
#int (d x)/(sqrt (x^2+4))=l n(x/2+sqrt(1+x^2/4))+C#
