\( \int{\frac{x^3dx}{x^2-9}} = \int{xdx} + \int{\frac{9xdx}{x^2-9}}\)

\(\frac{x}{(x+3)(x-3)} = \frac{A}{x+3} + \frac{B}{x-3}\)

\(x = A(x-3) + B (x+3)\)

If \(x = -3\), then \(A = \frac{1}{2}\)

If \(x = 3\), then \(B = \frac{1}{2}\)

\( \frac{x}{(x^2-9)} = \frac{1}{2(x+3)}+ \frac{1}{2(x-3)} \)

\(\int \frac{xdx}{(x^2-9)} = \int \frac{dx}{2(x+3)}+ \int \frac{dx}{2(x-3)}\)

\(\int \frac{xdx}{(x^2-9)} = \frac{1}{2} (\ln (x+3) + \ln(x-3))\)

\( \int{\frac{x^3dx}{x^2-9}} = \int{xdx} +9[\frac{1}{2} (\ln (x+3) + \ln(x-3))] \)

\( \int{\frac{x^3dx}{x^2-9}} = \frac{x^2}{2} +\frac{9}{2} (\ln (x+3) + \ln(x-3)) + C \)