Example 2.42. Compute the integral $I=\int_{0}^{1} \frac{x^{m}-x^{n}}{\ln x} d x$, $(m>0, n>0)$.
Solution. Let $F(m)=\int_{0}^{1} \frac{x^{m}-x^{n}}{\ln x} d x$, then the derivative

$$
\frac{d F(m)}{d m}=\int_{0}^{1} \frac{x^{m} \ln x d x}{\ln x}=\int_{0}^{1} x^{m} d x=\left.\frac{x^{m+1}}{m+1}\right|_{0} ^{1}=\frac{1}{m+1}
$$

Hence, $F(m)=\int_{0}^{m} \frac{d m}{m+1}=\ln |m+1|+C$, but for $m=$ $=n F(n)=0$, so $C=-\ln (n+1)$, thus,

$$
F(m)=\ln |m+1|-\ln |n+1|=\ln \left|\frac{m+1}{n+1}\right|
$$