change integrals by fubini's theorem can we use fubini here :
$$\int_{1}^{+\infty}\frac{|ln(t)|^p}{|1-t|^p} \int_{0}^{1/t}\frac{1}{x^{p-1}}dxdt$$ $$= $$ 
$$\int_{0}^{1/t}\frac{1}{x^{p-1}}\int_{1}^{+\infty}\frac{|ln(t)|^p}{|1-t|^p} dtdx$$
thanks

Since the integrand $$\frac{|\ln(t)|^p}{|1-t|^p x^{p-1}}$$ is nonnegative, you can use Fubini-Tonelli's theorem. But you did a mistake. If you want to integrate on the region $$\left\{(x,t)\,\middle|\,0\leq x \leq \tfrac{1}{t},\ 1\leq t\leq\infty\right\}$$ by changing the order of $x$ and $t$, you must observe that the above region is same as $$\left\{(x,t)\,\middle|\,1\leq t \leq \tfrac{1}{x},\ 0 \leq x \leq 1\right\}.$$ Then, the theorem tells us that $$\int_{1}^{\infty}\frac{|\ln(t)|^p}{|1-t|^p} \int_{0}^{1/t}\frac{1}{x^{p-1}}dxdt=\int_{0}^{1}\frac{1}{x^{p-1}}\int_{1}^{1/x}\frac{|\ln(t)|^p}{|1-t|^p} dtdx$$
