Calculate: integral of (3x+9)/((3-x) * (x^2+9)) x

Expression: $\int{ \frac{ 3x+9 }{ \left( 3-x \right) \times \left( {x}^{2}+9 \right) } } \mathrm{d} x$

Rewrite the fraction using partial-fraction decomposition

$\int{ \frac{ 1 }{ 3-x }+\frac{ x }{ {x}^{2}+9 } } \mathrm{d} x$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\int{ \frac{ 1 }{ 3-x } } \mathrm{d} x+\int{ \frac{ x }{ {x}^{2}+9 } } \mathrm{d} x$

Evaluate the indefinite integral

$-\ln\left({|3-x|}\right)+\int{ \frac{ x }{ {x}^{2}+9 } } \mathrm{d} x$

Evaluate the indefinite integral

$-\ln\left({|3-x|}\right)+\frac{ 1 }{ 2 } \times \ln\left({{x}^{2}+9}\right)$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }-\ln\left({|3-x|}\right)+\frac{ 1 }{ 2 } \times \ln\left({{x}^{2}+9}\right)+C,& C \in ℝ\end{array}$