Question

Интеграл x^3/(x^2+2x-1) dx

Answer 1

$displaystyle intfrac{x^3dx}{x^2+2x-1}=intfrac{(x^2+2x-1-2x+1)x}{x^2+2x-1}dx=\=int(x-frac{2x^2+4x-2-5x+2}{x^2+2x-1})dx=int(x-2+frac{5x-2}{x^2+2x-1})dx\int(x-2+frac{5}{2}frac{2x+2-frac{14}{5}}{x^2+2x-1})=int(x-2)dx+frac{5}{2}intfrac{d(x^2+2x-1)}{x^2+2x-1}-\-7intfrac{d(x+1)}{(x+1)^2-2}=frac{x^2}{2}-2x+frac{5}{2}ln|x^2+2x-1|-\-frac{7}{2sqrt2}ln|frac{x+1-sqrt2}{x+1+sqrt2}|+C$