Question

Evaluate: $\int \frac{dx}{\sqrt{x^2+x+1}}$

• A. $\ln |1+2x+2\sqrt{x^2+x+1}|+c$
• B. $-\ln |1+2x-2\sqrt{x^2+x+1}|+c$
• C. $-\ln |1-2x-2\sqrt{x^2+x+1}|+c$
• D. None of these

Answer 1

The correct option is A $\ln |1+2x+2\sqrt{x^2+x+1}|+c$

Let $I=\int \frac{dx}{\sqrt{x^2+x+1}}$
$I=\int \frac{dx}{\sqrt{x^2+2x\cdot \left(\frac{1}{2}\right)+\frac{1}{4}+1-\frac{1}{4}}}$
$I=\int \frac{dx}{\sqrt{{(x+\frac{1}{2})}^2-\frac{3}{4}}}$
$I=\int \frac{dx}{\sqrt{{(x+\frac{1}{2})}^2-{\left(\frac{\sqrt{3}}{2}\right)}^2}}$
$I=\ln |x+\frac{1}{2}+\sqrt{{(x+\frac{1}{2})}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}|+C_1$
$I=\ln |2x+1+2\sqrt{x^2+x+1}|+C$