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The image contains a mathematical problem written in Vietnamese. The primary problem is to find the value of \( y = \frac{1}{5} \) for which a certain integral expression equals zero. Let me transcribe and solve the problem for you.

The problem is as follows:

Given the integral equation:

\[
\int_0^1 (5x^3 - 2x^2 - 4x - 1) \, dx = 0
\]

You are to find the correct value of \( y \) for which the equation holds true.

The options provided are:

A. 5  
B. 1  
C. 7  
D. 8

### Let's solve the problem step by step.

### Step 1: Evaluate the Integral

First, we compute the definite integral:

\[
\int_0^1 (5x^3 - 2x^2 - 4x - 1) \, dx
\]

The integral of each term is:

\[
\int 5x^3 \, dx = \frac{5x^4}{4}
\]
\[
\int -2x^2 \, dx = -\frac{2x^3}{3}
\]
\[
\int -4x \, dx = -2x^2
\]
\[
\int -1 \, dx = -x
\]

So, the integral becomes:

\[
\left[\frac{5x^4}{4} - \frac{2x^3}{3} - 2x^2 - x \right]_0^1
\]

### Step 2: Evaluate at the Limits

Plug in the limits from 0 to 1:

At \( x = 1 \):

\[
\frac{5(1)^4}{4} - \frac{2(1)^3}{3} - 2(1)^2 - 1 = \frac{5}{4} - \frac{2}{3} - 2 - 1
\]

At \( x = 0 \), all terms will be zero.

So we get:

\[
\frac{5}{4} - \frac{2}{3} - 2 - 1
\]

Now let's calculate this:

\[
\frac{5}{4} = 1.25, \quad \frac{2}{3} \approx 0.67, \quad 2 = 2, \quad 1 = 1
\]

So the expression simplifies to:

\[
1.25 - 0.67 - 2 - 1 = -2.42
\]

### Step 3: Simplify the Expression

Since the problem asks when this expression equals zero, we need to check the correct value of \( y \) that could balance this expression to zero. Since the definite integral is negative, the correct choice should satisfy this.

Given the values, it looks like the problem involves a choice mechanism or possibly a mistake in the options.

### Final Check:

Given the context, if the problem is to choose among predefined values where the equation balances, none of these values seem to immediately make it zero, suggesting there might be another aspect or context (like a redefined range) to consider. 

**However, with the provided options, we would typically look for a standard approach (maybe even applying a substitution). Without an explicit answer, this would require more information.**

### Conclusion:

Given the complexity and the context of the problem, further clarification might be required, especially if dealing with integral rebalancing, scaling, or additional conditions applied. Let me know if you want further details or clarifications!

---

**Relative Questions:**

1. How does substitution affect the calculation of definite integrals?
2. What is the significance of the definite integral in physics or engineering?
3. Can you find a real-world example where definite integrals are used?
4. How does changing the limits of integration affect the outcome?
5. What strategies can be used when definite integrals result in negative values?

**Tip:** Always simplify your integral expressions as much as possible before evaluating them at the limits; this reduces the chance of errors.

Given the integral equation: 
\[ \int_0^1 (5x^3 - 2x^2 - 4x - 1) \, dx = 0 \]
Find the value of y = 1/5 that satisfies the equation.

Explore the step-by-step solution to find the value of y that satisfies the definite integral equation \( \int_0^1 (5x^3 - 2x^2 - 4x - 1) \, dx = 0 \). This problem highlights core calculus concepts and is suitable for students in Grades 11-12.

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