If `I=int(dx)/(x^(3)(x^(8)+1)^(3//4))=(lambda(1+x^(8))^((1)/(4)))/(x^(2))+c` (where c is the constant of integration), then the value of `lambda` is equal to

Options

A. 2
B. `(1)/(2)`
C. `-2`
D. `-(1)/(2)`

Answer

To solve the problem, we need to find the value of \( \lambda \) in the given integral equation:

\[ I = \int \frac{dx}{x^3 (x^8 + 1)^{3/4}} = \frac{\lambda (1 + x^8)^{1/4}}{x^2} + c \]

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form. We can factor out \( x^8 \) from \( (x^8 + 1)^{3/4} \):

\[ I = \int \frac{dx}{x^3 (x^8 + 1)^{3/4}} = \int \frac{dx}{x^3 \left( x^8 \left(1 + \frac{1}{x^8}\right) \right)^{3/4}} \]

### Step 2: Simplify the Integral
This can be simplified further:

\[ = \int \frac{dx}{x^3 \cdot x^{6} \cdot \left(1 + \frac{1}{x^8}\right)^{3/4}} = \int \frac{dx}{x^9 \left(1 + \frac{1}{x^8}\right)^{3/4}} \]

### Step 3: Substitute
Next, we can make a substitution to simplify the integral. Let:

\[ t = 1 + \frac{1}{x^8} \implies dt = -\frac{8}{x^9} dx \implies dx = -\frac{x^9}{8} dt \]

### Step 4: Change the Integral
Substituting \( dx \) into the integral gives:

\[ I = \int \frac{-\frac{x^9}{8} dt}{x^9 t^{3/4}} = -\frac{1}{8} \int \frac{dt}{t^{3/4}} \]

### Step 5: Integrate
Now we can integrate:

\[ -\frac{1}{8} \int t^{-3/4} dt = -\frac{1}{8} \cdot \frac{t^{1/4}}{1/4} + c = -\frac{1}{2} t^{1/4} + c \]

### Step 6: Substitute Back
Now substitute back \( t = 1 + \frac{1}{x^8} \):

\[ I = -\frac{1}{2} \left(1 + \frac{1}{x^8}\right)^{1/4} + c \]

### Step 7: Rewrite in Required Form
We can rewrite this as:

\[ I = -\frac{1}{2} \left(1 + x^8\right)^{1/4} \cdot \frac{1}{x^2} + c \]

### Step 8: Compare with Given Expression
Now we compare this with the given expression:

\[ \frac{\lambda (1 + x^8)^{1/4}}{x^2} + c \]

From this comparison, we can see that:

\[ \lambda = -\frac{1}{2} \]

### Conclusion
Thus, the value of \( \lambda \) is:

\[ \lambda = -\frac{1}{2} \]