4) $\int \frac{x^{2}+2 x}{x^{2}+2 x+1} d x$

Step 1 ：Simplify the function: \(f(x) = \frac{x^2 + 2x}{x^2 + 2x + 1}\)

Step 2 ：Find the integral: \(\int f(x) dx = \int \frac{x^2 + 2x}{x^2 + 2x + 1} dx = x + \frac{1}{x+1} + C\)

Step 3 ：Final Answer: \(\boxed{x + \frac{1}{x+1} + C}\)