Let $d, r, \alpha >0$, $j\ge 1$, $t_j\in\mathbb{R}$, $N\ge 2$ be a natural number with conjugate $N^{\prime}$ and $\omega_{N-1}$ the area of the unit sphere in $\mathbb{R}^N$. I am trying to justify the equality $$\int_{d/j}^d e^{\alpha t_j^{N^{\prime}}[\log(d/r)]^{N^{\prime}}/(\omega_{N-1}\log j)^{1/(N-1)}} r^{N-1} dr =d^N \log j\int_0^1 e^{-Nt[1-(t_j/t_0)^{N^{\prime}}t^{1/(N-1)}]\log j} dt, $$ where $t=\log(d/r)/ \log j$ and $t_0^{N^{\prime}}=[N\omega_{N-1}^{1/(N-1)}]/\alpha$.
Actually I do not understand how it is made the change of variables and how to change the extrema of integration. Could someone please help me to understand that?
Thank you in advance.