Calculate $\int_0^1\frac{x^9-1}{\log x}{\rm d}x$ using Beta and Gamma function

Question

I used Wolfram to find the answer which is $\log(10)$ but can't solve no matter how hard I try.

$$\int_0^1\frac{x^9-1}{\log x}{\rm d}x$$

Answer 1

$$I(a)=\int_{0}^{1} x^a dx=\frac{1}{1+a}.~~~~(1)$$ Integrate (1) w.r,t, $a$ $$\int_{0}^{1} \frac{x^a}{\ln x}=\ln(1+a)+C~~~(2)$$ Put $a=0$. then $$\int_{0}^{1} \frac{1}{\ln x} dx=C~~~~(3)$$ Using (3) in (2), we get $$\int_{0}^{1} \frac{x^a-1}{\ln x} dx=\ln(1+a)$$ Let $a=9$, to get $$\int_{0}^{1} \frac{x^9-1}{\ln x} dx=\ln(1+9)=\ln 10$$

Answer 2

Let $u=-\ln x$ so the integral is $\int_0^\infty\frac{e^{-u}-e^{-10u}}{u}du$, a Frullani integral equal to $\ln10$. I'm not aware of any solution in Beta and Gamma functions, and given the form of the answer it's unclear that one should exist.