Answer
$$\ln \left| {{x^4} - 10{x^2} + 9} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{4{x^3} - 20x}}{{{x^4} - 10{x^2} + 9}}} dx \cr
& {\text{Integrate by using the substitution method}} \cr
& \,\,\,{\text{Let }}u = {x^4} - 10{x^2} + 9,\,\,\,\,du = \left( {4{x^3} - 20x} \right)dx \cr
& \cr
& {\text{Write the integrand in terms of }}u \cr
& \int {\frac{{4{x^3} - 20x}}{{{x^4} - 10{x^2} + 9}}} dx = \int {\frac{{du}}{u}} \cr
& \cr
& {\text{Integrating}} \cr
& \int {\frac{{du}}{u}} = \ln \left| u \right| + C \cr
& {\text{Write in terms of }}x;{\text{ substitute }}{x^4} - 10{x^2} + 9{\text{ for }}u \cr
& = \ln \left| {{x^4} - 10{x^2} + 9} \right| + C \cr} $$
