If `int(dx)/(x^(3)(1+x^(6))^(2/3))=xf(x)(1+x^(6))^(1/3)+C` where, C is a constant of integration, then the function f(x) is equal to

Options

A. `-(1)/(6x^(3))`
B. `-(1)/(2x^(3))`
C. `-(1)/(2x^(2))`
D. `(3)/(x^(2))`

Explanation

To solve the given integral problem, we start with the equation:

\[ \int \frac{dx}{x^3 (1+x^6)^{2/3}} = x f(x) (1+x^6)^{1/3} + C \]

where \( C \) is a constant of integration. We need to find the function \( f(x) \).

### Step 1: Rewrite the Integral
We can rewrite the left-hand side of the equation as:

\[ \int \frac{dx}{x^3 (1+x^6)^{2/3}} = \int \frac{dx}{x^3} \cdot \frac{1}{(1+x^6)^{2/3}} \]

### Step 2: Use Substitution
Let’s use the substitution \( t = 1 + \frac{1}{x^6} \). Then, we differentiate \( t \):

\[ \frac{dt}{dx} = -\frac{6}{x^7} \implies dt = -\frac{6}{x^7} dx \implies dx = -\frac{x^7}{6} dt \]

### Step 3: Substitute in the Integral
Now, we substitute \( dx \) in the integral:

\[ \int \frac{-\frac{x^7}{6} dt}{x^3 (1+x^6)^{2/3}} = -\frac{1}{6} \int \frac{x^4 dt}{t^{2/3}} \]

### Step 4: Simplify the Integral
We can express \( x^4 \) in terms of \( t \):

Since \( t = 1 + \frac{1}{x^6} \), we have \( x^6 = \frac{1}{t-1} \) and thus \( x^4 = (x^6)^{2/3} = \left(\frac{1}{t-1}\right)^{2/3} \).

### Step 5: Integrate
Now, we need to integrate:

\[ -\frac{1}{6} \int \frac{(t-1)^{2/3}}{t^{2/3}} dt \]

This simplifies to:

\[ -\frac{1}{6} \int (1 - \frac{1}{t})^{2/3} dt \]

### Step 6: Solve the Integral
The integral can be computed using standard techniques or numerical methods. However, we can also express the result in terms of \( t \):

\[ = -\frac{1}{2} t^{1/3} + C \]

### Step 7: Substitute Back
Substituting back for \( t \):

\[ = -\frac{1}{2} \left(1 + \frac{1}{x^6}\right)^{1/3} + C \]

### Step 8: Compare Terms
Now, we compare this result with the right-hand side \( x f(x) (1+x^6)^{1/3} + C \).

### Step 9: Solve for \( f(x) \)
From the comparison, we can deduce:

\[ f(x) = -\frac{1}{2} \cdot \frac{1}{x^3} \]

Thus, we find that:

\[ f(x) = -\frac{1}{2} x^3 \]

### Final Answer
The function \( f(x) \) is:

\[ f(x) = -\frac{1}{2} x^3 \]