(1)

\begin{align*} \log\Gamma\left(x+1\right) & =\int_{0}^{x}\frac{d}{dx}\log\Gamma\left(x+1\right)dx\\ & =\int_{0}^{x}\psi\left(x+1\right)dx\\ & =\int_{0}^{x}\left\{ -\gamma+\sum_{k=1}^{\infty}(-1)^{k+1}\zeta(k+1)z^{k}\right\} dx\slug{jupvfrgr}\\ & =-\gamma x+\sum_{k=1}^{\infty}(-1)^{k+1}\frac{\zeta(k+1)}{k+1}z^{k+1}\\ & =-\gamma x+\sum_{k=2}^{\infty}\frac{(-1)^{k}\zeta\left(k\right)}{k}x^{k} \end{align*}

(1)-2

\begin{align*} \log\Gamma\left(x+1\right) & =\log\left(x\Gamma\left(x\right)\right)\\ & =\log x+\log\Gamma\left(x\right)\\ & =\log x-\log\left(xe^{\gamma x}\prod_{k=1}^{\infty}\left(1+\frac{x}{k}\right)e^{-\frac{x}{k}}\right)\\ & =\log x-\left(\log x+\gamma x+\sum_{k=1}^{\infty}\log\left(1+\frac{x}{k}\right)-\sum_{k=1}^{\infty}\frac{x}{k}\right)\\ & =\log x-\left(\log x+\gamma x+\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j}\left(\frac{x}{k}\right)^{j}-\sum_{k=1}^{\infty}\frac{x}{k}\right)\\ & =\log x-\left(\log x+\gamma x+\sum_{k=1}^{\infty}\sum_{j=2}^{\infty}\frac{(-1)^{j+1}}{j}\left(\frac{x}{k}\right)^{j}\right)\\ & =\log x-\left(\log x+\gamma x+\sum_{j=2}^{\infty}\frac{(-1)^{j+1}}{j}x^{j}\sum_{k=1}^{\infty}\frac{1}{k^{j}}\right)\\ & =\log x-\left(\log x+\gamma x+\sum_{j=2}^{\infty}\frac{(-1)^{j+1}}{j}x^{j}\zeta\left(j\right)\right)\\ & =-\gamma x+\sum_{j=2}^{\infty}\frac{(-1)^{j}\zeta\left(j\right)}{j}x^{j} \end{align*}

(2)

\begin{align*} \gamma & =\left[\gamma x\right]_{x=1}\\ & =\left[-\log\Gamma\left(1+x\right)+\sum_{k=2}^{\infty}\frac{(-1)^{k}\zeta\left(k\right)}{k}x^{k}\right]_{x=1}\\ & =\sum_{k=2}^{\infty}\frac{(-1)^{k}\zeta\left(k\right)}{k} \end{align*}