The value of `int1/(x^(2)(x^(4)+1)^(3//4))` dx is equal to

Options

A. `(1+1/x^(4))^(/1//4)` + C
B. `(x^(4)+1)^(1//4)+C`
C. `(1-1/x^(4))^(1//4)+C`
D. `-(1+1/x^(4))^(1//4)`+C

Answer

To solve the integral \( \int \frac{1}{x^2 (x^4 + 1)^{3/4}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{1}{x^2 (x^4 + 1)^{3/4}} \, dx \]

### Step 2: Simplify the Expression
We can rewrite the integral by factoring out \( x^4 \) from the term \( (x^4 + 1)^{3/4} \):
\[ I = \int \frac{1}{x^2 \left(x^4 \left(1 + \frac{1}{x^4}\right)\right)^{3/4}} \, dx \]
This simplifies to:
\[ I = \int \frac{1}{x^2 x^{3} \left(1 + \frac{1}{x^4}\right)^{3/4}} \, dx \]
\[ = \int \frac{1}{x^5 \left(1 + \frac{1}{x^4}\right)^{3/4}} \, dx \]

### Step 3: Substitution
Let \( t = 1 + \frac{1}{x^4} \). Then, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = -\frac{4}{x^5} \implies dx = -\frac{x^5}{4} dt \]

### Step 4: Substitute in the Integral
Substituting \( dx \) and \( t \) into the integral:
\[ I = \int \frac{-\frac{x^5}{4}}{x^5 t^{3/4}} \, dt \]
This simplifies to:
\[ I = -\frac{1}{4} \int t^{-3/4} \, dt \]

### Step 5: Integrate
Now we integrate:
\[ I = -\frac{1}{4} \cdot \frac{t^{1/4}}{1/4} + C = -t^{1/4} + C \]

### Step 6: Substitute Back
Now substitute back \( t = 1 + \frac{1}{x^4} \):
\[ I = -\left(1 + \frac{1}{x^4}\right)^{1/4} + C \]

### Final Answer
Thus, the value of the integral is:
\[ \int \frac{1}{x^2 (x^4 + 1)^{3/4}} \, dx = -\left(1 + \frac{1}{x^4}\right)^{1/4} + C \]