$\int \sqrt{x+2} d x$

Step 1 ：The integral of a function is the area under the curve of the function. In this case, we are asked to find the integral of the square root of x+2.

Step 2 ：This is a simple integral that can be solved using the power rule for integration, which states that the integral of x^n dx is (1/(n+1))x^(n+1).

Step 3 ：However, the function is not in the form of x^n, so we need to rewrite it in that form before we can apply the power rule. The square root of x+2 can be written as (x+2)^(1/2).

Step 4 ：Now we can apply the power rule. The integral of the function sqrt(x+2) is 2*(x + 2)^(3/2)/3.

Step 5 ：Final Answer: \(\boxed{\frac{2}{3}(x + 2)^{\frac{3}{2}}}\)