Question

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

• A. $\frac{2x+1}{4}\sqrt{x^2-x+1}+\frac{3}{8}\log \{\frac{2x+1}{2}+\sqrt{x^2-x+1}\}$
• B. $\frac{2x-1}{4}\sqrt{x^2-x+1}+\frac{3}{8}\log \{\frac{2x-1}{2}+\sqrt{x^2-x+1}\}$
• C. $\frac{2x+1}{2}\sqrt{x^2-x+1}+\frac{3}{8}\log \{\frac{2x-1}{4}+\sqrt{x^2-x+1}\}$
• D. $\frac{2x-1}{3}\sqrt{x^2-x+1}-\log \{\frac{2x-1}{2}+\sqrt{x^2-x+1}\}$
• E. $\frac{2x-1}{3}\sqrt{x^2-x+1}+\frac{3}{8}\log \{\frac{2x-1}{2}+\sqrt{x^2-x+1}\}$
• F. $\frac{2x-1}{2}\sqrt{x^2-x+1}+\frac{3}{8}\log \{\frac{2x-1}{4}+\sqrt{x^2-x+1}\}$
• G. $\frac{2x-1}{4}\sqrt{x^2-x+1}-\frac{3}{8}\log \{\frac{2x-1}{2}+\sqrt{x^2-x+1}\}$

Answer 1

The correct option is B $\frac{2x-1}{4}\sqrt{x^2-x+1}+\frac{3}{8}\log \{\frac{2x-1}{2}+\sqrt{x^2-x+1}\}$

$\int \sqrt{x^2-x+1}dx=\int \sqrt{{(x-\frac{1}{2})}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}dx$
$=\frac{x-\frac{1}{2}}{2}\sqrt{{(x-\frac{1}{2})}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}$

$+\frac{1}{2}{\left(\frac{\sqrt{3}}{2}\right)}^2\log \{(x-\frac{1}{2})+\sqrt{{(x-\frac{1}{2})}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}\}$
$=\frac{2x-1}{4}\sqrt{x^2-x+1}+\frac{3}{8}\log \{\frac{2x-1}{2}+\sqrt{x^2-x+1}\}$