Where can I find the first proof of $\int_0^1H_xdx=\gamma$

Question

We can prove $$\int_0^1H_xdx=\gamma$$ as such:$$\int_0^1H_xdx=\int_0^1\sum_{k=1}^\infty\left(\frac1k-\frac1{k+x}\right)dx=\sum_{k=1}^\infty\left(\frac1k-\ln\left(1+\frac1k\right)\right)=\gamma$$ where $H_x=\sum_{k=1}^\infty\left(\frac1k-\frac1{k+x}\right)$ is the generalization of the harmonic numbers. But I don't know where I can find a reference for this identity. When was it first made? I found this on a Wikipedia page for a list of integrals of Euler's constant without citation. It's the second to last one here. Another proof of this formula is also accepted.