Integrate (x+2)^2 dx

There are multiple methods that be applied here to reach a solution, integration by substitution or by applying the basic rules of integration. The first step would be to expand the expression within the parentheses and separate into multiple indefinite integrals. Next, the strategy would involve solving each individual integral (x^2, 4x, and 4 respectively) and obtaining x^3/3, 2x^2, and 4x. You would also add on an arbitary constant C as this is an indefinite integral.
The other strategy, integration by substitution, is less efficient in solving this problem but is very important when solving more complex integrals such as those involving trigonometric functions. For this problem, you would make x+2 = u or any variable you want. Thus, the integral would not be integrated with respect to x but instead u as well. To find the relationship between u and x, the next step would be to differentiate u = x + 2, giving du = 1 dx. Therefore, the new integral becomes u^2 du. The solution looks straightforward from here; u^3/3 + C. Here on, all you need to do is substitute x back in for u and expand the parentheses to obtain the answer (same as above).