`int(1)/(x(1+root(3)(x))^(2))dx`is equal to

Options

A. `3{"log"((x^(1//3))/(1+x^(1//3)))+(1)/(1+root(3)(x))}+C`
B. `3{"log"((x^(1//3))/(1+x^(1//3)))+(1)/(1+x}+C`
C. `3{"log"((x^(1//3))/(1+x^(1//3)))-(1)/(1+x^(1//3))}+C`
D. none of these

Explanation

To solve the integral \( \int \frac{1}{x(1+\sqrt[3]{x})^2} \, dx \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Substitution**: Let \( x = t^3 \). Then, differentiating both sides gives \( dx = 3t^2 \, dt \). Also, we have \( \sqrt[3]{x} = t \).

**Hint**: Use substitution to simplify the integral.

2. **Change of Variables**: Substitute \( x \) and \( dx \) in the integral:
\[ \int \frac{1}{t^3(1+t)^2} \cdot 3t^2 \, dt = 3 \int \frac{t^2}{t^3(1+t)^2} \, dt = 3 \int \frac{1}{t(1+t)^2} \, dt \]

**Hint**: Simplifying the integral after substitution can help in reducing the complexity.

3. **Partial Fraction Decomposition**: We can express \( \frac{1}{t(1+t)^2} \) in terms of partial fractions:
\[ \frac{1}{t(1+t)^2} = \frac{A}{t} + \frac{B}{1+t} + \frac{C}{(1+t)^2} \]
Multiplying through by the denominator \( t(1+t)^2 \) gives:
\[ 1 = A(1+t)^2 + Bt(1+t) + Ct \]

**Hint**: Set up the equation for partial fractions carefully to find the constants.

4. **Finding Coefficients**: Expanding the right-hand side:
\[ 1 = A(1 + 2t + t^2) + Bt + Bt^2 + Ct \]
Collecting like terms:
\[ 1 = (A)t^2 + (2A + B + C)t + A \]
By comparing coefficients, we get:
- \( A = 1 \)
- \( 2A + B + C = 0 \)
- \( A = 0 \)

From \( A = 1 \), we find \( B + C = -2 \) and \( B + C = -2 \). Solving these gives \( B = -1 \) and \( C = -1 \).

**Hint**: Use the method of comparing coefficients to solve for the unknowns.

5. **Rewrite the Integral**: Now we can rewrite the integral:
\[ 3 \int \left( \frac{1}{t} - \frac{1}{1+t} - \frac{1}{(1+t)^2} \right) dt \]

**Hint**: Breaking down the integral into simpler parts makes it easier to integrate.

6. **Integrate Each Term**:
\[ 3 \left( \ln |t| - \ln |1+t| + \frac{1}{1+t} \right) + C \]

**Hint**: Remember to integrate each term separately.

7. **Substitute Back**: Replace \( t \) with \( \sqrt[3]{x} \):
\[ 3 \left( \ln |\sqrt[3]{x}| - \ln |1+\sqrt[3]{x}| + \frac{1}{1+\sqrt[3]{x}} \right) + C \]

**Hint**: Always substitute back to the original variable to express the final answer.

8. **Final Simplification**: The final answer can be simplified as:
\[ \ln |x| - 3 \ln |1+\sqrt[3]{x}| + \frac{3}{1+\sqrt[3]{x}} + C \]

**Hint**: Combine logarithmic terms using properties of logarithms if necessary.

### Final Answer:
\[ \int \frac{1}{x(1+\sqrt[3]{x})^2} \, dx = 3 \left( \ln |\sqrt[3]{x}| - \ln |1+\sqrt[3]{x}| + \frac{1}{1+\sqrt[3]{x}} \right) + C \]