use partial fractions to getOriginally Posted bynavynukejones

$\displaystyle \int \frac{5x^2-x+11}{(x^2+4)(x-1)}dx=\int \left( \frac{2x+1}{x^2+4}+\frac{3}{x-1}\right)dx=$

now we use termise division of the first term to get

$\displaystyle \int \left( \frac{2x}{x^2+4}+\frac{1}{x^2+4}+\frac{3}{x-1}\right)dx=\ln|x^2+4|+\frac{1}{2}\tan^{-1}(x/2)+3\ln|x-1|+C$