Answer
$$\frac{2}{5}\ln \left| x \right| + \frac{3}{5}\ln \left| {x + 5} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{x + 2}}{{{x^2} + 5x}}} dx \cr
& {\text{Decompose the integrand into partial fractions}} \cr
& \frac{{x + 2}}{{{x^2} + 5x}} = \frac{{x + 2}}{{x\left( {x + 5} \right)}} \cr
& \frac{{x + 2}}{{x\left( {x + 5} \right)}} = \frac{A}{x} + \frac{B}{{x + 5}} \cr
& x + 2 = A\left( {x + 5} \right) + Bx \cr
& {\text{For }}x = 0 \cr
& 2 = A\left( 5 \right) + B\left( 0 \right) \to A = \frac{2}{5} \cr
& {\text{For }}x = - 5 \cr
& - 5 + 2 = A\left( 0 \right) + B\left( { - 5} \right) \to B = \frac{3}{5} \cr
& \frac{{x + 2}}{{x\left( {x + 5} \right)}} = \frac{{2/5}}{x} + \frac{{3/5}}{{x + 5}} \cr
& \int {\frac{{x + 2}}{{{x^2} + 5x}}} dx = \int {\left( {\frac{{2/5}}{x} + \frac{{3/5}}{{x + 5}}} \right)} dx \cr
& {\text{Integrate}} \cr
& {\text{ = }}\frac{2}{5}\ln \left| x \right| + \frac{3}{5}\ln \left| {x + 5} \right| + C \cr} $$
