To solve this problem using $u$-substitution, let’s proceed through each component:

1. Choose $u$ for substitution: Let: $u = 3x - 4$

2. Differentiate $u$ with respect to $x$ to find $\frac{du}{dx}$: $\frac{du}{dx} = 3 \Rightarrow dx = \frac{du}{3}$

3. Determine the new limits of integration: When $x = 1$: $u = 3(1) - 4 = -1$

   When $x = 4$: $u = 3(4) - 4 = 8$

4. Rewrite the integral: Substitute $u = 3x - 4$, $dx = \frac{du}{3}$, and change the limits from $x = 1$ to $x = 4$ to $u = -1$ to $u = 8$: $\int_1^4 (3x - 4)^4 \, dx = \int_{-1}^{8} u^4 \cdot \frac{du}{3} = \frac{1}{3} \int_{-1}^{8} u^4 \, du$

5. Evaluate the integral: Now we integrate $\frac{1}{3} \int_{-1}^{8} u^4 \, du$: $\frac{1}{3} \cdot \left[ \frac{u^5}{5} \right]_{-1}^{8} = \frac{1}{3} \cdot \left( \frac{8^5}{5} - \frac{(-1)^5}{5} \right)$

6. Calculate $8^5$ and $(-1)^5$:

   • $8^5 = 32768$
   • $(-1)^5 = -1$

   Substitute these values: $= \frac{1}{3} \cdot \left( \frac{32768}{5} + \frac{1}{5} \right) = \frac{1}{3} \cdot \frac{32769}{5} = \frac{32769}{15}$

Filling the Boxes

• $u = 3x - 4$
• $\frac{du}{dx} = 3 \Rightarrow dx = \frac{du}{3}$
• $a = -1$
• $b = 8$
• $f(u) = \frac{1}{3} u^4$
• The value of the original integral is $\frac{32769}{15}$.

Would you like further explanation on any step or have additional questions?

Here are five questions that expand on this topic:

1. How does changing the limits of integration help in $u$-substitution?
2. Why is the constant $\frac{1}{3}$ factored out in the final integral?
3. What are some common functions where $u$-substitution is particularly useful?
4. How would this integral change if the exponent were different?
5. Can we use a different substitution method for this integral, such as integration by parts?

Tip: Always verify the new limits after substitution to avoid mistakes in integration boundaries.