"U = (1 + x) and integrate" is a mathematical expression that represents a function, "U", which is equal to the sum of 1 and x, and the process of finding the area under the curve of this function.
To integrate U = (1 + x), you can use the power rule for integration, which states that the integral of x^n is equal to (x^(n+1))/(n+1). In this case, n is equal to 1, so the integral of x is (x^2)/2. Therefore, the integral of U = (1 + x) is (1/2)x^2 + C, where C is a constant.
The purpose of integrating U = (1 + x) is to find the area under the curve of this function. This is useful in many applications, such as in physics, engineering, and economics, where finding the total amount of something is important.
For example, if we want to find the area under the curve of U = (1 + x) from x = 0 to x = 2, we can use the integral formula to calculate it. The integral of U = (1 + x) is (1/2)x^2 + C, so the area under the curve is ((1/2)2^2 + C) - ((1/2)0^2 + C) = 2 + C, where C is the constant of integration.
No, the integral of U = (1 + x) is not the same as the derivative of (1 + x). The integral is the reverse process of taking the derivative, so while the derivative of (1 + x) is equal to 1, the integral of U = (1 + x) is (1/2)x^2 + C. They are related, but not the same.