Which of the following is true for $\int_{1}^{0} x\ln x\, \text dx$?

Question

Which of the following is true for $\int_{1}^{0} x\ln x\,\text dx$

1. it is equal to $−1/4$
2. it is divergent
3. it is equal to an irrational number
4. does not have a closed form
5. it is impossible to evaluate this integral

According to my calculations it evaluates to $\frac{2x^{2}\ln x - x^{2}}{4}$
No when I put the values for limit 0, it then comes out to be $-\infty$ for $\log x$
Then which of the options are correct?

Answer 1

$\int_1^0 x\ ln(x)\ dx = \frac{x^2}{2}ln(x)|_1^0 - \int_1^0 \frac{x^2}{2}\frac{1}{x}dx= +\frac{1}{4}$