Step 1: Use Integration by Parts
To evaluate the integral \(\int_2^8 t^2 \ln(2t) dt\), we will use the integration by parts formula, which is given by:
\[
\int u \, dv = uv - \int v \, du
\]
Here, we choose:
\[
u = \ln(2t) \quad \text{and} \quad dv = t^2 \, dt
\]
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