If `int (dx)/(x ^(4) (1+x^(3))^2)=a ln |(1+x ^(3))/(x ^(3))| +(b)/(x ^(3)) +(c)/(1+ x^(2)) +d.` then (where d is arbitrary constant)

Options

A. `a =1/3 , b =1/3 , c =1/3`
B. `a= 2/3, b=-1/3, c =1/3`
C. `a =2/3, b =-1/3 , c=-1/3`
D. `a = 2/3b =1/3, c=-1/3`

Answer

To solve the integral

\[ \int \frac{dx}{x^4 (1+x^3)^2} \]

we will use substitution and partial fractions.

### Step 1: Rewrite the Integral
We can rewrite the integral as follows:

\[ \int \frac{dx}{x^4 (1+x^3)^2} = \int \frac{1}{x^4} \cdot \frac{1}{(1+x^3)^2} \, dx \]

### Step 2: Use Substitution
Let \( t = 1 + x^3 \). Then, we have:

\[ dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2} \]

Also, from our substitution, we can express \( x^3 \) in terms of \( t \):

\[ x^3 = t - 1 \quad \Rightarrow \quad x = (t - 1)^{1/3} \]

Now, we need to express \( x^4 \) in terms of \( t \):

\[ x^4 = (x^3)^{4/3} = (t - 1)^{4/3} \]

### Step 3: Substitute in the Integral
Now substituting these into the integral gives:

\[ \int \frac{1}{(t - 1)^{4/3} t^2} \cdot \frac{dt}{3x^2} \]

We also need to express \( x^2 \):

\[ x^2 = (t - 1)^{2/3} \]

Thus, the integral becomes:

\[ \int \frac{1}{(t - 1)^{4/3} t^2} \cdot \frac{dt}{3(t - 1)^{2/3}} = \frac{1}{3} \int \frac{1}{(t - 1)^2 t^2} dt \]

### Step 4: Partial Fraction Decomposition
Now we need to perform partial fraction decomposition on

\[ \frac{1}{(t - 1)^2 t^2} \]

Assuming:

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

Multiplying through by the denominator \((t - 1)^2 t^2\) gives:

\[ 1 = A(t - 1)^2 + B(t - 1)^2 + Ct^2(t - 1) + Dt^2 \]

### Step 5: Solve for Coefficients
To find the coefficients \(A\), \(B\), \(C\), and \(D\), we can substitute convenient values for \(t\):

1. Set \(t = 0\) to find \(B\).
2. Set \(t = 1\) to find \(D\).
3. Set other values to find \(A\) and \(C\).

After solving these equations, we will find the values of \(A\), \(B\), \(C\), and \(D\).

### Step 6: Integrate Each Term
Once we have the partial fractions, we can integrate each term separately:

\[ \int \left( \frac{A}{t} + \frac{B}{t^2} + \frac{C}{t - 1} + \frac{D}{(t - 1)^2} \right) dt \]

### Step 7: Substitute Back
After integrating, we will substitute back \(t = 1 + x^3\) to express the integral in terms of \(x\).

### Final Expression
The final result will be of the form:

\[ a \ln \left| \frac{1 + x^3}{x^3} \right| + \frac{b}{x^3} + \frac{c}{1 + x^2} + d \]

### Step 8: Determine Constants
By matching coefficients from the integrated expression with the given form, we can determine the values of \(a\), \(b\), and \(c\).