If ∫x+1x2+x+1dxx2+x+1=kx−1x2+x+1+C then the value of k

If ∫x+1×2+x+1dxx2+x+1=kx−1×2+x+1+C then the value of k

If $\int \frac{x + 1}{(x^{2} + x + 1)} \frac{d x}{\sqrt{x^{2} + x + 1}} = k \frac{x - 1}{\sqrt{x^{2} + x + 1}} + C$ then the value of $k$

Differentiating both the sides w.r.t.x, and simplifying , we get

$\frac{x + 1}{{(x^{2} + x + 1)}^{3 / 2}} = \frac{3 k}{2} \frac{x + 1}{{(x^{2} + x + 1)}^{3 / 2}} \Rightarrow k = \frac{2}{3}$