The integration of x with respect to x is a fundamental concept in calculus. The integral of x is expressed mathematically as:
∫ x dx = (1/2)x² + C
Here, C represents the constant of integration, which is included because integration can yield a family of functions that differ by a constant.
To understand why this is the case, let’s consider the concept of integration as finding the area under a curve. In this case, the curve is represented by the function y = x, which is a straight line that goes through the origin with a slope of 1. When we integrate to find the area under this line from a certain point to another, we are essentially accumulating the values of x over that interval.
The formula (1/2)x² comes from the power rule in integration. The power rule states that the integral of x^n is (1/(n + 1))x^(n + 1), where n is a real number. For the case of integrating x, we have n = 1. Applying the power rule:
∫ x^1 dx = (1/(1 + 1))x^(1 + 1) = (1/2)x²
This explains the (1/2)x² component of the result. Therefore, whenever you see the integral of x, remember it relates to finding the area under the curve of the linear function y = x, and the result includes a constant to account for the indefinite nature of integration.