`int(1)/(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 \( I = \int \frac{1}{x^2 (x^4 + 1)^{3/4}} \, dx \), we will follow these steps:

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

### Step 2: Factor Out \( x^4 \)
Notice that we can factor \( x^4 \) from the term \( (x^4 + 1) \):
\[ I = \int \frac{1}{x^2} \cdot \frac{1}{x^{3} (1 + \frac{1}{x^4})^{3/4}} \, dx \]
This simplifies to:
\[ I = \int \frac{1}{x^5 (1 + \frac{1}{x^4})^{3/4}} \, dx \]

### Step 3: Substitution
Let \( t = 1 + \frac{1}{x^4} \). Then, we differentiate \( t \):
\[ dt = -\frac{4}{x^5} \, dx \quad \Rightarrow \quad dx = -\frac{1}{4} x^5 \, dt \]
From our substitution, we can express \( x^5 \) in terms of \( t \):
\[ x^5 = \frac{-1}{4} dt \]

### Step 4: Substitute in the Integral
Now substitute \( t \) into the integral:
\[ I = \int \frac{-1/4}{(t)^{3/4}} \cdot \left(-\frac{1}{4} dt\right) \]
This simplifies to:
\[ I = \frac{1}{4} \int t^{-3/4} \, dt \]

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

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

### Final Answer
Thus, the final result of the integral is:
\[ I = \left(1 + \frac{1}{x^4}\right)^{1/4} + C \]

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