How do you integrate ##ln(x^2-x+2)dx##?

##=(x- 1/2) ln(x^2-x+2) - 2x + sqrt 7 arctan ( (2x-1)/sqrt 7)##

##int dx qquad ln(x^2-x+2)##

we use IBP

##int u v' = uv - int u' v##

and the trick here is

##u = ln(x^2-x+2), u' = (2x-1)/(x^2-x+2)## ##v' = 1, v = x##

so we have

##xln(x^2-x+2) - int dx qquad x(2x-1)/(x^2-x+2) qquad star##

for ##int dx qquad x(2x-1)/(x^2-x+2) ## this becomes:

##int dx qquad (2x^2-x)/(x^2-x+2) ##

next bit is like long division only easier

##= int dx qquad (2x^2-2x + 4 + x- 4)/(x^2-x+2) ##

##= int dx qquad 2 + (x- 4)/(x^2-x+2) ##

now we set it up for a log solution by setting up this pattern: ##(f'(x)) / f(x)##

##= 2x + int dx qquad (1/2(2x- 8))/(x^2-x+2) ##

##=2x + int dx qquad (1/2(2x- 1)-7/2)/(x^2-x+2) ##

##= 2x + int dx qquad (1/2(2x- 1))/(x^2-x+2) - 7/2 1/(x^2-x+2) ##

##= 2x + 1/2 ln(x^2-x+2) - 7/2 int dx qquad 1/(x^2-x+2) ##

if we plug this all into ##star## we have

##xln(x^2-x+2) - ( 2x + 1/2 ln(x^2-x+2) - 7/2 int dx qquad 1/(x^2-x+2) )##

##=(x- 1/2) ln(x^2-x+2) - 2x + 7/2 int dx qquad 1/(x^2-x+2) qquad square##

for ## int dx qquad 1/(x^2-x+2) ##

we complete the square, looking for a tan sub to finish it off, so

## int dx qquad 1/((x-1/2)^2 - 1/4 + 2) ##

##= int dx qquad 1/((x-1/2)^2 + 7/4) ##

##= int dx qquad 1/(7/4 tan^2 phi + 7/4) ## using the sub ##(x-1/2)^2 = 7/4 tan^2 phi## or ##(x-1/2) = sqrt 7/2 tan phi##

so ##dx = sqrt 7/2 sec^2 phi \ d phi##

##=sqrt 7/2 int d phi qquad sec^2 phi \ 1/(7/4 sec^2 phi ) ##

##=2/ sqrt 7 int d phi qquad ##

##=2/ sqrt 7 arctan (2 (x-1/2)/sqrt 7)##

parking this back into ##square##

##=(x- 1/2) ln(x^2-x+2) - 2x + 7/2 2/ sqrt 7 arctan (2 (x-1/2)/sqrt 7)##

##=(x- 1/2) ln(x^2-x+2) - 2x + sqrt 7 arctan ( (2x-1)/sqrt 7)##