Evaluate the following integral: ∫ log((1/x )- 1)dx, x ∈ [0, 1]

Question

Evaluate the following integral:

\(\int\limits_{0}^{1}log(\cfrac{1}{\text x}-1)d\text x \)

Answer 1

Let I = \(\int\limits_{0}^{1}log(\cfrac{1}{\text x}-1)d\text x \)

I = \(\int\limits_{0}^{1}log\cfrac{(1-\text x)}{\text x}d\text x \)

We know

thus

By property, we know that

Adding equation 2 and equation 3 we have

Thus

I = 0