Evaluate the Integral $\int(x+1)^{3} d x$

Step 1 ：The integral of a function can be found using the power rule for integration, which states that the integral of x^n dx is (1/(n+1))x^(n+1) + C, where C is the constant of integration.

Step 2 ：In this case, the function to be integrated is (x+1)^3. We can apply the power rule directly to this function.

Step 3 ：The integral of the function (x+1)^3 is x^4/4 + x^3 + 3*x^2/2 + x. This is the antiderivative of the function, and represents the area under the curve of the function from x = 0 to x. The constant of integration is not included in this expression, as it is assumed to be 0 for the purposes of this calculation.

Step 4 ：Final Answer: The integral of \((x+1)^{3}\) with respect to x is \(\boxed{\frac{x^{4}}{4} + x^{3} + \frac{3x^{2}}{2} + x}\).