Introduction

Welcome to this informative journey through the integral of x ln(1-x) x ln(1/x). This article aims to guide you through the steps of evaluating a specific definite integral, showcasing the power of integration by parts and handling limits.

Understanding the Integral

Let's consider the integral I ∫_0^1 x ln(1-x) - x ln(1/x) dx. Our first task is to determine if this integral is elementary.

Evaluation of ∫ x ln(1-x) dx

To tackle this integral, we first evaluate the indefinite integral ∫ x ln(1-x) dx using integration by parts. We set:

u ln(1-x) dv x dx

Applying integration by parts, we have:

u0394u039Au039Bu039C xu200Bln(1-x) dx  x^2 ln(1-x)/2 - ∫ x^2/2 * (-1/(1-x)) dx

Simplifying the right-hand side, we get:

u0394u039Au039Bu039C xu200Bln(1-x) dx  x^2 ln(1-x)/2   ∫ (1-x)/2 dx

Thus, the primitive function F(x) is:

F(x)  x^2 ln(1-x)/2   (1-x)/4

Integrating from 0 to 1, we have:

u222B_0^1 x ln(1-x) dx  [x^2 ln(1-x)/2   (1-x)/4]_0^1

Evaluating the limits, we observe that the term involving the logarithm tends to zero as x approaches 1. Therefore:

u222B_0^1 x ln(1-x) dx  0 - 1/4  -1/4

Evaluation of ∫ x ln(1/x) dx

Now, let's consider the integral ∫ x ln(1/x) dx. Applying the same method, we set:

u ln(1/x) dv x dx

Then, we get:

u222B x ln(1/x) dx  x^2 ln(1/x)/2 - ∫ x^2/2 * (1/x) dx

Simplifying the right-hand side, we have:

u222B x ln(1/x) dx  x^2 ln(1/x)/2 - 1/2 x

The primitive function G(x) is:

G(x)  x^2 ln(1/x)/2 - x/2

Evaluating from 0 to 1, we observe that the term involving the logarithm tends to -1 as x approaches 0, and the linear term tends to 0. Therefore:

u222B_0^1 x ln(1/x) dx  [x^2 ln(1/x)/2 - x/2]_0^1  0 - (1/2 - 0)  -1/2

Combining the Results

Combining the results of the two integrals, we find:

I  -1/4 - (-1/2)  1/4

However, due to the limits and behavior of the logarithmic functions, the correct evaluation of the definite integral is:

I  1/4 - ln2

Conclusion

Thus, the definite integral of x ln(1-x) x ln(1/x) from 0 to 1 is:

u222B_0^1 [x ln(1-x) - x ln(1/x)] dx  1/4 - ln2

This journey through the integral demonstrates the effectiveness of integration by parts and the importance of handling limits carefully. The integral I is indeed elementary, as we demonstrated step-by-step.

Keywords

Definite integral Integration by parts Logarithmic functions