用重积分求解定积分

问题: 求

$$I=\int_{0}^{1}\frac{x-1}{\ln x}\ud x.$$

提示

$$I=\int_{0}^{1}\int_{0}^{1}x^{y}\ud y\ud x=\int_{0}^{1}\int_{0}^{1}x^{y}\ud x\ud y=\int_{0}^{1}\frac{1}{y+1}\ud y=\ln(1+y)\mid_{0}^{1}=\ln2.$$