Understanding the Integration of log(1-x)/(1-x) with Limits from -1 to 1

Let's delve into an interesting problem in calculus where we examine the logarithmic function and its properties. We will see how understanding the properties of odd functions can simplify the process of integration significantly.

Introduction to the Logarithmic Function

Consider the function f(x) log#x2061;1 - x1 - x defined for -1 x 1.

Analysis of the Function

Let's first understand the behavior of the function at negative values. Specifically, we are interested in the value of the function when x 0. We start by substituting x rarr; -x into the function:

Transformation of the Function

We have the function f-x log#x2061;1 - -x1 - x. Simplifying the expression inside the logarithm, we get:

f-x log#x2061;1 x1 - x -log#x2061;1 - x1 x -f(x)

Note that the second equality follows from the fact that the argument of the logarithm is the reciprocal of the original, and the third equality follows from the property that the logarithm of a reciprocal is the negative of the logarithm of the original value.

Odd Functions and Symmetry

By transforming the function, we can see that f-x-f(x). This means that the function f(x) is an odd function. Odd functions exhibit symmetry about the origin, a property that will prove crucial for our integration.

Application of Symmetry in Integration

Given that the function is odd, the integral over a symmetric interval will simplify. Specifically, we are interested in the integral from -1 to 1. The integral can be expressed as:

#x222B; -1 1 f(x) dx #x222B; -1 0 f(x) dx #x222B; 0 1 f(x) dx

Using the property of odd functions, we can rewrite the integral as:

#x222B; -1 1 f(x) dx - #x222B; 1 0 f(x) dx #x222B; 0 1 f(x) dx #x222B; 0 1 ( f(x) - f(x) ) dx 0

The integral of an odd function over a symmetric interval is always zero. Therefore, the integral of f(x) from -1 to 1 is zero.

Conclusion

In conclusion, the problem of integrating the function f(x) over the interval [-1, 1] is simplified by recognizing the function's odd nature. This symmetry about the origin allows us to conclude that the integral is zero without explicitly computing the integral. The analysis of the function's behavior and properties of odd functions have provided a succinct and elegant solution to the problem.

Key Takeaways

Understanding the properties of odd functions can greatly simplify the integration process. Integral of an odd function over a symmetric interval is always zero. Symmetry about the origin is a powerful tool in calculus.