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integration of (x+1)/sqrt(x^2-x+1)

integration of (x+1)/sqrt(x^2-x+1)

Grade:Upto college level

2 Answers

Sukhendra Reddy Rompally B.Tech Mining Machinery Engg, ISM Dhanbad
93 Points
10 years ago
Dear student, write the numerator as (2x-1)+3,then you can divide the integrand into 2 parts,1st one is a simple 1/root x form nd the second-write the denominator as root of (x-1/2)^2 + 3/4.i guess you can solve it from here.if you still have a problem,feel free to call me on 07209736303
vikas askiitian expert
509 Points
10 years ago

(x^2-x+1)=x^2-x+1/4+3/4

            =(x-1/2)^2 + 3/4

 integral(x+1)/sqrt(x^2-x+1)=x+1/sqrt{(x-1/2)^2 + 3/4}

 now put x-1/2=t

             dx=dt

 integral(t+3/2)/sqrt(t^2+3/4)

       = int{ t/sqrt(t^2+3/4)  + 3/4sqrt(t^2+3/4) }

       = sqrt(t^2 +3/4)   + 3/4log{sqrt(t^2+3/4) +t} +c

        =sqrt{ (x-1/2)^2 +3/4}   +   3/4 log{sqrt(x-1/2)^2 +x +1/4 } +c

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