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Let's break down the given integral using **u-substitution**. The integral is:

\[
\int_1^4 (5x - 3)^5 \, dx
\]

We are asked to apply the method of **u-substitution** where the general form is:

\[
\int_a^b f(g(x))g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du
\]

### Step 1: Choosing \( u \)

To simplify the integral, we let:
\[
u = 5x - 3
\]

### Step 2: Differentiating \( u \)

Next, we differentiate \( u \) with respect to \( x \) to find \( du \):

\[
du = 5 \, dx \quad \text{or equivalently} \quad dx = \frac{du}{5}
\]

### Step 3: Changing the limits of integration

The original limits of integration are from \( x = 1 \) to \( x = 4 \). We need to convert these limits in terms of \( u \).

- When \( x = 1 \): 
  \[
  u = 5(1) - 3 = 2
  \]
- When \( x = 4 \): 
  \[
  u = 5(4) - 3 = 17
  \]

So the limits of integration for \( u \) are from 2 to 17.

### Step 4: Substituting into the integral

The integral becomes:

\[
\int_2^{17} u^5 \cdot \frac{du}{5}
\]

Simplifying:

\[
\frac{1}{5} \int_2^{17} u^5 \, du
\]

### Step 5: Solving the integral

Now we integrate \( u^5 \):

\[
\int u^5 \, du = \frac{u^6}{6}
\]

Thus, the integral becomes:

\[
\frac{1}{5} \cdot \left[ \frac{u^6}{6} \right]_2^{17}
\]

### Step 6: Applying the limits

Substitute \( u = 17 \) and \( u = 2 \):

\[
\frac{1}{5} \cdot \left( \frac{17^6}{6} - \frac{2^6}{6} \right)
\]

Now, let's compute the result. I will calculate this.The value of the original integral is:

\[
804583.5
\]

Here’s a summary of the u-substitution steps:

- \( u = 5x - 3 \)
- \( du = 5 \, dx \), so \( dx = \frac{du}{5} \)
- Limits for \( u \) go from 2 to 17
- The integral simplifies to \( \frac{1}{5} \int_2^{17} u^5 \, du \)
- The computed result is \( 804583.5 \)

Would you like further clarification on any of the steps, or do you have additional questions?

### 5 Related Questions:
1. How does u-substitution simplify complex integrals?
2. Why do we need to adjust the limits when performing u-substitution?
3. Can u-substitution be applied to any integral?
4. What happens if you forget to change the limits during substitution?
5. What is the significance of the constant factor when substituting \( du \)?

### Tip:
Always check if you need to adjust the limits of integration after making a substitution. This is a common point where errors can occur in definite integrals.

Using the method of u-substitution, evaluate the integral ∫ from 1 to 4 of (5x - 3)^5 dx.

Learn how to use the method of u-substitution to solve the integral of (5x - 3)^5 from 1 to 4. This step-by-step guide covers the substitution process, changing limits, and calculating the final value.

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