If `int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b` , then a is equal

Options

A. `1//3`
B. `2//3`
C. `-1//3`
D. `-2//3`

Explanation

To solve the integral

\[ I = \int \frac{1}{x \sqrt{1 - x^3}} \, dx \]

we will follow these steps:

### Step 1: Multiply and Divide by \(x^2\)

We can rewrite the integral by multiplying and dividing by \(x^2\):

\[ I = \int \frac{x^2}{x^3 \sqrt{1 - x^3}} \, dx \]

### Step 2: Substitute \(1 - x^3 = y^2\)

Let \(1 - x^3 = y^2\). Then, differentiating both sides gives:

\[ -3x^2 \, dx = 2y \, dy \quad \Rightarrow \quad x^2 \, dx = -\frac{2y}{3} \, dy \]

### Step 3: Express \(x^3\) in terms of \(y\)

From the substitution, we have:

\[ x^3 = 1 - y^2 \]

### Step 4: Substitute into the Integral

Now, substituting \(x^2 \, dx\) and \(x^3\) into the integral, we get:

\[ I = \int \frac{-\frac{2y}{3} \, dy}{(1 - y^2) \sqrt{y^2}} = -\frac{2}{3} \int \frac{y \, dy}{(1 - y^2) y} = -\frac{2}{3} \int \frac{1}{1 - y^2} \, dy \]

### Step 5: Simplify the Integral

The integral simplifies to:

\[ I = -\frac{2}{3} \int \frac{1}{1 - y^2} \, dy \]

### Step 6: Use Partial Fraction Decomposition

We can express \(\frac{1}{1 - y^2}\) as:

\[ \frac{1}{1 - y^2} = \frac{1}{(y - 1)(y + 1)} = \frac{A}{y - 1} + \frac{B}{y + 1} \]

Solving for \(A\) and \(B\) gives:

\[ A + B = 0 \quad \text{and} \quad A - B = 1 \]

From these equations, we find \(A = \frac{1}{2}\) and \(B = -\frac{1}{2}\).

### Step 7: Integrate

Now we can rewrite the integral:

\[ I = -\frac{2}{3} \left( \frac{1}{2} \int \frac{1}{y - 1} \, dy - \frac{1}{2} \int \frac{1}{y + 1} \, dy \right) \]

Integrating gives:

\[ I = -\frac{2}{3} \left( \frac{1}{2} \log |y - 1| - \frac{1}{2} \log |y + 1| \right) + C \]

This simplifies to:

\[ I = -\frac{1}{3} \log \left| \frac{y - 1}{y + 1} \right| + C \]

### Step 8: Substitute Back for \(y\)

Substituting back \(y = \sqrt{1 - x^3}\):

\[ I = -\frac{1}{3} \log \left| \frac{\sqrt{1 - x^3} - 1}{\sqrt{1 - x^3} + 1} \right| + C \]

### Step 9: Compare with Given Expression

The given expression is:

\[ I = a \log \left| \frac{\sqrt{1 - x^3} - 1}{\sqrt{1 - x^3} + 1} \right| + b \]

By comparing, we find:

\[ a = -\frac{1}{3} \]

### Conclusion

Thus, the value of \(a\) is:

\[ \boxed{-\frac{1}{3}} \]