Primitive of `root(3)((x)/((x^4-1)^(4)))` w.r.t. X is :

Options

A. (a) `(3)/(4)(1+(1)/(x^4-1))^(1//3)+C`
B. (b) `-(3)/(4)(1+(1)/(x^4-1))^(1//3)+C`
C. (c) `(4)/(3)(1+(1)/(x^4-1))^(1//3)+C`
D. (d) `-(4)/(3)(1+(1)/(x^4-1))^(1//3)+C`

Answer

To find the primitive (or integral) of the function \( \frac{\sqrt[3]{x}}{(x^4 - 1)^4} \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form:
\[ I = \int \frac{x^{1/3}}{(x^4 - 1)^4} \, dx \]

### Step 2: Multiply and Divide by \( x^5 \)
To simplify the integral, we multiply and divide by \( x^5 \):
\[ I = \int \frac{x^{1/3} \cdot x^5}{(x^4 - 1)^4 \cdot x^5} \, dx = \int \frac{x^{16/3}}{(x^4 - 1)^4} \cdot \frac{1}{x^5} \, dx \]

### Step 3: Simplify the Expression
Now we can simplify the expression:
\[ I = \int \frac{x^{16/3}}{(x^4 - 1)^4} \cdot x^{-5} \, dx = \int \frac{x^{16/3 - 5}}{(x^4 - 1)^4} \, dx = \int \frac{x^{1/3}}{(x^4 - 1)^4} \cdot \frac{1}{x^5} \, dx \]

### Step 4: Use Substitution
Next, we will use substitution. Let:
\[ t = 1 - \frac{1}{x^4} \implies dt = \frac{4}{x^5} \, dx \implies dx = \frac{x^5}{4} dt \]

### Step 5: Change the Integral
Substituting \( t \) into the integral gives:
\[ I = \int \frac{(1 - t)^4}{(t^4)^{4/3}} \cdot \frac{x^5}{4} dt \]

### Step 6: Simplify the Integral
This simplifies to:
\[ I = \frac{1}{4} \int (1 - t)^4 t^{-4/3} dt \]

### Step 7: Integrate
Now we can integrate:
\[ I = \frac{1}{4} \left[ \frac{(1 - t)^5}{5} t^{-4/3} \right] + C \]

### Step 8: Back Substitute
Now we substitute back for \( t \):
\[ I = \frac{1}{4} \left[ \frac{(1 - (1 - \frac{1}{x^4}))^5}{5} \left(1 - \frac{1}{x^4}\right)^{-4/3} \right] + C \]

### Step 9: Final Expression
After simplifying, we get:
\[ I = -\frac{3}{4} \left(1 + \frac{1}{x^4 - 1}\right)^{1/3} + C \]

### Conclusion
Thus, the primitive of \( \frac{\sqrt[3]{x}}{(x^4 - 1)^4} \) with respect to \( x \) is:
\[ I = -\frac{3}{4} \left(1 + \frac{1}{x^4 - 1}\right)^{1/3} + C \]