#int (2x-5)/(x^2-4x+5)#
#2x-5=2x-4-1#
#int (2x-5)/(x^2-4x+5)=int (2x-4-1)/(x^2-4x+5)d x#
#"split the integration"#
#int (2x-5)/(x^2-4x+5)=color(red)(int (2x-4)/(x^2-4x+5)d x)-color(green)(int1/(x^2-4x+5)d x) #
#color(red)(int (2x-4)/(x^2-4x+5)d x)#
#"Substitute "u=x^2-4x+5" ; " d u=2x-4#
#color(red)(int (2x-4)/(x^2-4x+5)d x)=int (d u)/u=l n u#
#"Undo substitution"#
#color(red)(int (2x-4)/(x^2-4x+5)d x)=l n(|x^2-4x+5|)#
#color(green)(int1/(x^2-4x+5)d x)=#
#x^2-4x+5=(x-2)^2+1#
#"please remember that "int (d x)/(x^2+a^2)=1/a arc tan (x/a)+C#
#color(green)(int1/(x^2-4x+5)d x)=int (d x)/((x-2)^2+1)=arc tan (x-2)#
#int (2x-5)/(x^2-4x+5)=l n(|x^2-4x+5|-arc tan(x-2))+C#