1. The problem is to evaluate the integral $$\int \frac{x}{x+a} \, dx$$. 2. We use the method of algebraic manipulation to simplify the integrand before integrating. 3. Rewrite the integrand by dividing numerator and denominator: $$\frac{x}{x+a} = \frac{(x+a)-a}{x+a} = \frac{x+a}{x+a} - \frac{a}{x+a} = 1 - \frac{a}{x+a}$$. 4. So the integral becomes $$\int \left(1 - \frac{a}{x+a}\right) dx = \int 1 \, dx - a \int \frac{1}{x+a} \, dx$$. 5. Integrate each term separately: $$\int 1 \, dx = x$$ and $$\int \frac{1}{x+a} \, dx = \ln|x+a|$$. 6. Therefore, the integral is $$x - a \ln|x+a| + C$$ where $C$ is the constant of integration. 7. Final answer: $$\boxed{x - a \ln|x+a| + C}$$.