To solve the given integral $\int 4(2 - 5x)^7 \, dx$ using $u$-substitution, we proceed as follows:

Step 1: Choose the substitution

Let: $u = 2 - 5x$ Then, the derivative of $u$ is: $\frac{du}{dx} = -5 \quad \text{or equivalently} \quad du = -5 \, dx \quad \text{or} \quad dx = -\frac{1}{5} \, du.$

Step 2: Rewrite the integral in terms of $u$

Substituting $u = 2 - 5x$ and $dx = -\frac{1}{5} \, du$, the integral becomes: $\int 4(2 - 5x)^7 \, dx = \int 4u^7 \left(-\frac{1}{5}\right) \, du = -\frac{4}{5} \int u^7 \, du.$

Step 3: Solve the integral

The integral of $u^7$ is: $\int u^7 \, du = \frac{u^8}{8}.$ Thus: $-\frac{4}{5} \int u^7 \, du = -\frac{4}{5} \cdot \frac{u^8}{8} = -\frac{4}{40} u^8 = -\frac{1}{10} u^8.$

Step 4: Back-substitute $u = 2 - 5x$

Substituting back $u = 2 - 5x$, the solution becomes: $-\frac{1}{10} (2 - 5x)^8 + C,$ where $C$ is the constant of integration.

Final Answer:

$\boxed{-\frac{1}{10} (2 - 5x)^8 + C}$

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Related Questions:

1. What is $u$-substitution, and when is it commonly used in integration?
2. How can you verify the correctness of the integral by differentiating the result?
3. How does the chain rule relate to $u$-substitution in integrals?
4. What happens if you choose a different substitution? Will the result still be correct?
5. How can you handle definite integrals using $u$-substitution?

Tip:

When using $u$-substitution, always double-check the substitution for $dx$ and ensure all terms in the integral are expressed in terms of $u$.