Evaluate the nine rational antiderivatives listed below: \[ \frac{x^2}{x^2+1},\quad \frac{4x^2+x+7}{x^2+1},\quad \frac{3x^2+x+3}{x(x^2+1)},\quad \frac{2x-3}{x^3+x}, \] \[ \frac{4x^2-11x-19}{(x-5)(x^2+1)},\quad \frac{3x^2+2x-7}{(x-5)(x^2+1)}, \] \[ \frac{-2x+4}{(x^2+1)(x-1)^2},\quad \frac{x^4+2x^3+5x^2+4x+6}{x^3+x^2+x+1}, \] and \[ \frac{2x^5+4x^3+4x}{x^4+1}. \]
Solution
The corresponding primitives are, respectively, \[ \boxed{x-\arctan x+C,} \] \[ \boxed{4x+\frac12\ln(x^2+1)+3\arctan x+C,} \] \[ \boxed{3\ln|x|+\arctan x+C,} \] \[ \boxed{-3\ln|x|+\frac32\ln(x^2+1)+2\arctan x+C,} \] \[ \boxed{\ln|x-5|+\frac32\ln(x^2+1)+4\arctan x+C,} \] \[ \boxed{3\ln|x-5|+2\arctan x+C,} \] \[ \boxed{-2\ln|x-1|-\frac1{x-1}+\ln(x^2+1)+\arctan x+C,} \] \[ \boxed{\frac12x^2+x+3\ln|x+1|+2\arctan x+C,} \] and \[ \boxed{x^2+\ln(x^4+1)+\arctan(x^2)+C.} \] Each follows from the partial-fraction decomposition of its integrand; differentiating each displayed primitive recovers the corresponding rational integrand.