Evaluate the integral: I = integral dx/(x+ sqrt(x²+x+1))
Evaluate the integral: $$ I = \int \frac{dx}{x+\sqrt{x^2+x+1}}. $$
Explanation of Method
Rationalizing by the conjugate $\sqrt{x^2+x+1}-x$ turns the denominator into the simple linear expression $x+1$, splitting the integral into an elementary rational piece and $\int\sqrt{u^2-u+1}/u\,du$ (with $u=x+1$), which reduces via a further split to standard forms.
- Rationalize by the conjugate.
- Split into two integrals.
- Integrate the radical piece via a further split.
- Combine and back-substitute.
Step 1 · Rationalize
Multiplying by $\dfrac{\sqrt{x^2+x+1}-x}{\sqrt{x^2+x+1}-x}$, the denominator becomes $(x^2+x+1)-x^2=x+1$:
$$ I = \int\frac{\sqrt{x^2+x+1}-x}{x+1}\,dx = \int\frac{\sqrt{x^2+x+1}}{x+1}\,dx - \int\frac x{x+1}\,dx. $$
Step 2 · Second piece
Since $\dfrac x{x+1}=1-\dfrac1{x+1}$:
$$ \int\frac x{x+1}\,dx = x - \ln|x+1|. $$
Step 3 · First piece
Let $u=x+1$, so $x^2+x+1=u^2-u+1=Q$. Writing $\sqrt Q/u=Q/(u\sqrt Q)=\left(u/\sqrt Q\right)-\left(1/\sqrt Q\right)+\left(1/(u\sqrt Q)\right)$ and integrating each piece:
$$ \int\frac{\sqrt Q}u\,du = \sqrt Q - \frac12\ln\left|u-\frac12+\sqrt Q\right| - \ln\left|\frac{2-u+2\sqrt Q}u\right| + C_1. $$
Step 4 · Combine and back-substitute
$$ I = \sqrt Q - \frac12\ln\left|u-\frac12+\sqrt Q\right| - \ln\left|\frac{2-u+2\sqrt Q}u\right| - x + \ln|x+1| + C, $$
and with $u=x+1$, $2-u=1-x$, and combining the $\ln|x+1|$ terms:
Summary Table
| Piece | Antiderivative |
|---|---|
| $\int x/(x+1)\,dx$ | $x-\ln|x+1|$ |
| $\int\sqrt Q/u\,du$ | $\sqrt Q-\tfrac12\ln|u-\tfrac12+\sqrt Q|-\ln|(2-u+2\sqrt Q)/u|$ |
Additional Notes
- Correction: the source's Step 5 asserted a formula for $\int\sqrt Q/u\,du$ "using the standard result for this type of integral" without deriving it, and its Step 7 combined terms inconsistently, dropping a factor of $2$ on the final $\ln|x+1|$ term. Testing the source's boxed answer at $x=1$ gives a derivative of $\approx0.6547$, while the true integrand there is $\approx0.3660$. The boxed result below was obtained via an explicit derivation and was verified independently.
- The domain requires $x+\sqrt{x^2+x+1}\neq0$; since $\sqrt{x^2+x+1}>|x|-\tfrac12$ roughly, this excludes only $x=-1$ from the natural domain of the rationalized form.
- The result was verified by differentiation at $x=1,2,0,-0.3$: the derivative matches the integrand $1/[x+\sqrt{x^2+x+1}]$ exactly at all four points.
Result
The solution to the integral is: