Question

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

Answer 1

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

$I=\frac{1}{2}\int (x+\frac{3}{2}+\frac{7}{4(x-\frac{1}{2})})dx$

$I=\frac{1}{2}(\frac{x^2}{2}+\frac{3x}{2}+\frac{7\ln (x-\frac{1}{2})}{4})+C$

Hence, this is the answer.