`int(1)/(xsqrt(1-x^(3)))`dx is equal to

Options

A. `(1)/(3)log|(sqrt(1-x^(3)-1))/(sqrt(1-x^(3))+1)|+C`
B. `(1)/(2)log|(sqrt(1-x^(2))+1)/(sqrt1-x^(2))-1|+C`
C. `(1)/(3)log|(1)/(sqrt(1-x^(3)))|+C`
D. none of these

Explanation

To solve the integral \( \int \frac{1}{x \sqrt{1 - x^3}} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{1}{x \sqrt{1 - x^3}} \, dx \]

### Step 2: Multiply and Divide by \(3x^2\)
To facilitate the integration, we multiply and divide the integrand by \(3x^2\):
\[ I = \int \frac{3x^2}{3x^3 \sqrt{1 - x^3}} \, dx \]

### Step 3: Substitute \(u = 1 - x^3\)
Let \(u = 1 - x^3\). Then, we differentiate:
\[ du = -3x^2 \, dx \quad \Rightarrow \quad dx = -\frac{du}{3x^2} \]

### Step 4: Substitute in the Integral
Substituting \(u\) and \(dx\) into the integral gives:
\[ I = \int \frac{3x^2}{3x^3 \sqrt{u}} \left(-\frac{du}{3x^2}\right) \]
This simplifies to:
\[ I = -\int \frac{1}{x^3 \sqrt{u}} \, du \]

### Step 5: Express \(x\) in terms of \(u\)
From the substitution \(u = 1 - x^3\), we have:
\[ x^3 = 1 - u \quad \Rightarrow \quad x = (1 - u)^{1/3} \]
Thus, \(x^3 = 1 - u\) implies:
\[ I = -\int \frac{1}{(1-u) \sqrt{u}} \, du \]

### Step 6: Simplify the Integral
Now we can simplify the integral:
\[ I = -\int \frac{1}{(1-u) \sqrt{u}} \, du \]

### Step 7: Use Partial Fraction Decomposition
To integrate this, we can use partial fractions:
\[ \frac{1}{(1-u) \sqrt{u}} = \frac{A}{1-u} + \frac{B}{\sqrt{u}} \]
Finding constants \(A\) and \(B\) will allow us to integrate each term separately.

### Step 8: Integrate Each Term
After finding \(A\) and \(B\), we can integrate:
\[ \int \frac{A}{1-u} \, du + \int \frac{B}{\sqrt{u}} \, du \]

### Step 9: Back Substitute
Finally, we back substitute \(u = 1 - x^3\) to express the integral in terms of \(x\).

### Final Answer
The final result will be in the form:
\[ I = -\frac{1}{3} \log(\sqrt{1 - x^3}) + C \]
where \(C\) is the constant of integration.

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