To integrate the expression $u' (u^2) \, du$ over the interval $[0, 5]$, we first need to clarify the expression. The prime notation $u'$ typically denotes a derivative, but within the integral as written, it seems you intend to integrate a function involving $u$.

Assuming $u'$ does not apply in this context, it looks like the expression to integrate could be either $u^2$ or $2u \cdot u'$. However, if we assume the correct interpretation is simply to integrate $u^2$ (since typically $u'$ would imply differentiation, not multiplication within the context of integration with respect to $u$), then the integral is:

$\int_0^5 u^2 \, du.$

Solution:

1. Indefinite Integral:

   The indefinite integral of $u^2$ is:

   $\int u^2 \, du = \frac{u^3}{3} + C.$

2. Definite Integral:

   Evaluate the integral from 0 to 5:

   $\int_0^5 u^2 \, du = \left[ \frac{u^3}{3} \right]_0^5.$

   Evaluate at the bounds:

   $\left[ \frac{u^3}{3} \right]_0^5 = \frac{5^3}{3} - \frac{0^3}{3} = \frac{125}{3} - 0 = \frac{125}{3}.$

Final Answer:

The integral of $u^2$ from 0 to 5 is:

$\frac{125}{3}.$

Please confirm the expression to ensure this interpretation is correct or provide additional context if necessary.