Evaluating a Definite Integral Involving Logarithms and Polynomial Terms

What is the result of the definite integral I ∫01 x ln(1-x) - x ln(1/x) dx? In this detailed step-by-step guide, we will explore the evaluation of this integral and demonstrate that such an integral, despite its complexity, is indeed an elementary one. The process involves integration by parts and careful manipulation of terms.

Step-by-Step Solution

First, let's break down the integral I into more manageable pieces. We start with the indefinite integral of x ln(1-x) and then apply the result to our original definite integral.

Evaluating ∫ x ln(1-x) dx

We perform integration by parts, letting:

This gives us:

Thus, we have:

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

Further simplification yields:

(1/2)x^2 ln(1-x) (1/4)∫ (x^2/(1-x)) dx

We can simplify the second integral as follows:

(1/2)x^2 ln(1-x) (1/4)∫ ((1-(1-x)^2)/(1-x)) dx

This yields:

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

Thus, the primitive of x ln(1-x) is:

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

Evaluating the Definite Integral

Plugging the limits into our primitive function:

(1/2)[1^2 ln(1-1) - 1^2/4] - (1/2)[0^2 ln(1-0) - 0^2/4] 0 - (1/4)

Observe that the term (1/2)[1^2 ln(1-1)] causes an issue as ln(0) is undefined. We need to evaluate the limit as x approaches 1:

limx→1 (1-x) ln(1-x) limx→1 -ln(x-1)/(1/(1-x)) 0 (using L'Hopital's Rule)

Therefore, our integral simplifies to:

-1/8 - 1/2 * 2ln2 - 1/4 -1/4 - ln2 1/4 -1/4 - ln2 1/4 -ln2 1/4

Hence, the value of the definite integral I is:

[boxed{1/4 - ln2}]

Conclusion

Through careful manipulation and integration by parts, we demonstrated that the given integral is indeed an elementary one. This process showcases the power of integration techniques in handling seemingly complex functions.