Question

  fraction numerator x squared plus x plus 1 over denominator x minus 1 end fraction

Answer 1

$$\dfrac{x^2+x+1}{x-1}=\dfrac{x^2-x+2x-2+3}{x-1}$$

$$=x+2+\dfrac{3}{x-1}, x\not=1$$

$$\dfrac{x^2+x+1}{x-1}=x+2+\dfrac{3}{x-1}, x\not=1$$

$$\int\dfrac{x^2+x+1}{x-1}dx=\int(x+2+\dfrac{3}{x-1})dx$$

$$=\dfrac{x^2}{2}+2x+3\ln(|x-1|)+C$$

$$\int\dfrac{x^2+x+1}{x-1}dx=\dfrac{x^2}{2}+2x+3\ln(|x-1|)+C$$