If `I=int x^(-11) (1+x^4)^(-1//2) \ dx`

Answer

To solve the integral \( I = \int x^{-11} (1 + x^4)^{-1/2} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int x^{-11} (1 + x^4)^{-1/2} \, dx \]
We can rewrite \( x^{-11} \) as \( \frac{1}{x^{11}} \) and \( (1 + x^4)^{-1/2} \) remains as is.

### Step 2: Change of Variables
Next, we can simplify the expression by making a substitution. Let:
\[ t = 1 + x^4 \]
Then, differentiating both sides gives:
\[ dt = 4x^3 \, dx \quad \Rightarrow \quad dx = \frac{dt}{4x^3} \]

### Step 3: Express \( x \) in Terms of \( t \)
From our substitution, we can express \( x^4 \) in terms of \( t \):
\[ x^4 = t - 1 \quad \Rightarrow \quad x = (t - 1)^{1/4} \]
Thus,
\[ x^3 = (t - 1)^{3/4} \]

### Step 4: Substitute into the Integral
Now we substitute \( dx \) and \( x^3 \) into the integral:
\[ I = \int \frac{1}{(t - 1)^{11/4}} (t)^{-1/2} \cdot \frac{dt}{4(t - 1)^{3/4}} \]
This simplifies to:
\[ I = \frac{1}{4} \int t^{-1/2} (t - 1)^{-11/4} \, dt \]

### Step 5: Simplify the Integral
Now we can simplify the integral:
\[ I = \frac{1}{4} \int t^{-1/2} (t - 1)^{-11/4} \, dt \]

### Step 6: Use the Beta Function
This integral can be evaluated using the Beta function or by recognizing it as a standard form. The integral can be expressed as:
\[ I = \frac{1}{4} B\left(\frac{1}{2}, \frac{1}{4}\right) \]
where \( B(x,y) \) is the Beta function.

### Step 7: Final Result
The result of the Beta function can be computed, but for the purposes of this solution, we can leave it in terms of the Beta function:
\[ I = \frac{1}{4} B\left(\frac{1}{2}, \frac{1}{4}\right) + C \]
where \( C \) is the constant of integration.