If I understood well the change of the limits of the integral when we make a substitution, it must looks like this :

Say we have $\displaystyle \int_0^1 3(3x+5)^4dx.$ Let $\displaystyle u=3x+5$, then $\displaystyle du=3dx$. So we have $\displaystyle \int_5^8 u^4du=\frac{u^5}{5} \big| _5^8$. I changed the limits of the integral of $\displaystyle 0$ and $\displaystyle 1$ to $\displaystyle 5$ and $\displaystyle 8$ because when $\displaystyle x=0$, $\displaystyle 3x+5=5$, etc. But now comes the step I don't understand : $\displaystyle \int_5^8 u^4du=\frac{u^5}{5} \big| _5^8=\frac{(3x+5)^5}{5}\big| _0^1$. Why the limits change again? It seems totally worthless to bother to change the limits of the integral if at last we change them as they were! I don't understand the why of we change the limits in the last step. (I understand why in the first step, it's because of the chain rule).