# Evaluation of $\int\frac{dx}{x+ \sqrt{x^2-x+1}}$

Evaluate :

$$\frac{dx}{x+ \sqrt{x^2-x+1}}$$

After dividing and multiply with $x-\sqrt{x^2-x+1}$, I get

$x+\ln |x-1|-\int \frac{\sqrt{x^2-x+1}}{x-1}dx$.

Is $\int \frac{\sqrt{x^2-x+1}}{x-1}dx$ is integrable in terms of elementary functions?

• Have you learned hyperbolic substitution?? Jan 21 '17 at 3:48

We have $$I =\int \frac {\sqrt {x^2-x+1}}{x-1} dx =\frac {1}{2 }\int \frac {\sqrt {(2x-1)^2+3}}{x-1} dx$$ Substituting $u =2x-1$, we get, $$I =\frac {1}{2} \int \frac {\sqrt{u^2+3}}{u-1} du$$ Now perform hyperbolic substitution $u =\sqrt {3} \operatorname {sinh}(v)$ and simplifying gives us $$I =\frac {3}{2} \int \frac {(\operatorname {cosh }(v))^2}{\sqrt {3}\operatorname{sinh}(v) -1} dv$$ which can be solved by Weierstrass substitution. Hope it helps.
There is a substitution, called the Euler substitution, which solves your integral rather neatly. In your case we set $t = \sqrt{x^2 - x + 1} + x$; after subtracting $x$ from both sides and squaring, we get $x^2 -x + 1 = t^2 - 2xt + x^2$, which can be solved (after subtracting $x^2$ from both sides) to get $x = \frac{t^2 - 1}{2t - 1}$. In this case, $\frac{dt}{dx} = \frac{1}{\sqrt{x^2 - x + 1}}(2x-1) + 1$, and so
• $\frac{dt}{dx}=\frac1{\color{red}{2}\sqrt{x^2-x+1}}(2x-1)+1$ Oct 18 '20 at 17:16