To evaluate the definite integral

$\int_{0}^{3} (8t - 1)^2 \, dt,$

we can expand the expression $(8t - 1)^2$ and integrate term-by-term.

Step-by-Step Solution:

1. Expand $(8t - 1)^2$:

   $(8t - 1)^2 = (8t)^2 - 2 \cdot 8t \cdot 1 + 1^2 = 64t^2 - 16t + 1.$

2. Rewrite the Integral:

   Substitute this expanded form into the integral:

   $\int_{0}^{3} (8t - 1)^2 \, dt = \int_{0}^{3} (64t^2 - 16t + 1) \, dt.$

3. Integrate Each Term Separately:

   Now, integrate term-by-term:

   $\int 64t^2 \, dt = 64 \cdot \frac{t^3}{3} = \frac{64t^3}{3},$ $\int -16t \, dt = -16 \cdot \frac{t^2}{2} = -8t^2,$ $\int 1 \, dt = t.$

   So, we get:

   $\int_{0}^{3} (64t^2 - 16t + 1) \, dt = \left[ \frac{64t^3}{3} - 8t^2 + t \right]_{0}^{3}.$

4. Evaluate the Integral at the Bounds:

   Substitute $t = 3$ and $t = 0$:

   $\left( \frac{64 \cdot 3^3}{3} - 8 \cdot 3^2 + 3 \right) - \left( \frac{64 \cdot 0^3}{3} - 8 \cdot 0^2 + 0 \right).$

   Simplify each term:

   • For $t = 3$: $\frac{64 \cdot 27}{3} - 8 \cdot 9 + 3 = \frac{1728}{3} - 72 + 3 = 576 - 72 + 3 = 507.$

   • For $t = 0$: $0.$

   So, the result is:

   $507.$

Final Answer:

$\int_{0}^{3} (8t - 1)^2 \, dt = 507.$

Would you like further details on any steps, or have any questions?

Here are five related questions for further practice:

1. How would you evaluate $\int_{0}^{3} (3t - 2)^2 \, dt$?
2. What is the process to evaluate $\int_{0}^{2} (5t + 3)^2 \, dt$?
3. Can you integrate $\int_{1}^{4} (2t + 1)^2 \, dt$?
4. How would changing the bounds affect $\int_{0}^{2} (8t - 1)^2 \, dt$?
5. What if the integrand was $(8t - 1)^3$ instead of $(8t - 1)^2$?

Tip: When integrating a squared expression, expanding it can often simplify the process before applying integration.