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Al solucionar la integral  usando el método de fracciones parciales, se obtiene:

a. ∫2x/(x(x-3)2) dx=((Ax+B)/x)+(C/(x-3)2)
b. ∫2x/(x(x-3)2) dx=(A/x)+(B/(x-3))+(C/(x-3)2)
c. ∫2x/(x(x-3)2)dx=(A/(x-3))+(C/(x+3))
d. ∫2x/(x(x-3)2) dx=(A/x)+(B/(x-3))


To solve the given integral using the method of partial fractions, we need to express the integrand as a sum of simpler fractions.

The integral given is:

\[
\int \frac{2x}{x(x-3)^2} \, dx
\]

To apply the method of partial fractions, the integrand \(\frac{2x}{x(x-3)^2}\) should be decomposed into simpler terms. The standard approach for such a decomposition is:

\[
\frac{2x}{x(x-3)^2} = \frac{A}{x} + \frac{B}{x-3} + \frac{C}{(x-3)^2}
\]

Here, we are expressing the fraction as a sum of simpler fractions where the denominators are the factors of the original denominator \(x(x-3)^2\).

Let's confirm this decomposition:

1. Multiply both sides by the common denominator \(x(x-3)^2\) to clear the fractions:

\[
2x = A(x-3)^2 + Bx(x-3) + Cx
\]

2. Expand and simplify the right side:

\[
2x = A(x^2 - 6x + 9) + Bx(x - 3) + Cx
\]
\[
2x = A(x^2 - 6x + 9) + B(x^2 - 3x) + Cx
\]

3. Combine like terms on the right side:

\[
2x = (A + B)x^2 + (-6A - 3B + C)x + 9A
\]

For the equation to hold true for all \(x\), the coefficients of corresponding powers of \(x\) on both sides must be equal.

- For \(x^2\): \(A + B = 0\)
- For \(x^1\): \(-6A - 3B + C = 2\)
- For the constant term: \(9A = 0\) (which implies \(A = 0\))

Substituting \(A = 0\) into the other equations:

- \(B = 0\) (since \(A + B = 0\))
- \(C = 2\) (from \(-6(0) - 3(0) + C = 2\))

Thus, the partial fraction decomposition is:

\[
\frac{2x}{x(x-3)^2} = \frac{0}{x} + \frac{0}{x-3} + \frac{2}{(x-3)^2} = \frac{2}{(x-3)^2}
\]

The correct answer from the given options is:

**Option b:**
\[
\int \frac{2x}{x(x-3)^2} \, dx = \frac{A}{x} + \frac{B}{x-3} + \frac{C}{(x-3)^2}
\]

where \(A = 0\), \(B = 0\), and \(C = 2\).

Let me know if you need further details or have any questions. 

Here are 8 related questions:

1. What is the method of partial fractions used for in integration?
2. How do you determine the coefficients in a partial fraction decomposition?
3. What conditions are necessary for the partial fraction decomposition to be valid?
4. Can partial fractions be used for non-rational functions? If not, why?
5. What other methods can be used to integrate rational functions?
6. How do you integrate a function like \(\frac{2}{(x-3)^2}\)?
7. What are the key differences between proper and improper fractions in this context?
8. How would the solution differ if the original integral had a higher power in the denominator?

**Tip:** Always check if the denominator can be factored fully when applying partial fractions, as this will guide you in setting up the correct decomposition.

Learn how to solve the integral ∫(2x / (x(x-3)^2)) dx using partial fractions. Step-by-step solution with detailed decomposition and verification. Suitable for advanced high school students and college-level learners.

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