$$ \int \frac { 4 x } { x ^ { 5 } - 2 } d x $$

Question

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It seems that the integral $$ \int \frac{4x}{x^5 - 2} \, dx $$ cannot be directly computed using the available tools. However, I can guide you through the process of solving it step by step.

### Step 1: Identify the Integral
We need to evaluate the integral:
$$
I = \int \frac{4x}{x^5 - 2} \, dx
$$

### Step 2: Use Substitution
A common technique for integrals of this form is to use substitution. Let's set:
$$
u = x^5 - 2
$$
Then, we differentiate $$u$$ with respect to $$x$$:
$$
\frac{du}{dx} = 5x^4 \quad \Rightarrow \quad du = 5x^4 \, dx \quad \Rightarrow \quad dx = \frac{du}{5x^4}
$$

### Step 3: Rewrite the Integral
Now, we can express $$x$$ in terms of $$u$$:
$$
x^4 = (u + 2)^{4/5}
$$
Substituting $$dx$$ and $$u$$ into the integral gives:
$$
I = \int \frac{4x}{u} \cdot \frac{du}{5x^4}
$$
This simplifies to:
$$
I = \frac{4}{5} \int \frac{1}{u} \, du
$$

### Step 4: Integrate
The integral of $$\frac{1}{u}$$ is:
$$
\int \frac{1}{u} \, du = \ln |u| + C
$$
Thus, we have:
$$
I = \frac{4}{5} \ln |u| + C
$$

### Step 5: Substitute Back
Now, substitute back $$u = x^5 - 2$$:
$$
I = \frac{4}{5} \ln |x^5 - 2| + C
$$

### Final Answer
The final result of the integral is:
$$
\int \frac{4x}{x^5 - 2} \, dx = \frac{4}{5} \ln |x^5 - 2| + C
$$

This is the solution to the integral. If you have any further questions or need additional assistance, feel free to ask!
The integral $$ \int \frac{4x}{x^5 - 2} \, dx $$ equals $$ \frac{4}{5} \ln |x^5 - 2| + C $$.

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Solución

Step 1: Identify the Integral

Step 2: Use Substitution

Step 3: Rewrite the Integral

Step 4: Integrate

Step 5: Substitute Back

Final Answer

Beyond the Answer

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