-3/2dx / (x^2+x+1)

Question

what is the integral of 3/2dx / (x^2+x+1)

Answer 1

Let J=-3/2∫dx/(x²+x+1)=

-3/2∫dx/(x²+x+¼+¾)=

-3/2∫dx/((x+½)²+¾).

Let x+½=(√3/2)tanθ, (x+½)²=¾tan²θ, (x+½)²+¾=¾tan²θ+¾=¾sec²θ;

dx=(√3/2)sec²θdθ; θ=tan^-1{(2x+1)/√3}

J=-¾√3∫sec²θdθ/[¾sec²θ]=-√3∫dθ=-√3θ+C, where C is integration constant.

J=-√3tan^-1{(2x+1)/√3}+C.